- Kitchen countertops: 36" standard. Tall people may prefer 37-38". Short people 32-35". To find the ideal countertop height, hold your elbows out at a 45-degree angle as if you were placing your hands on a counter. The distance from your hands to the floor is the ideal countertop height for you. However, beware that most appliances designed for 36" counter height.
- Counter stools: 24".
- Kitchen bar: 42-43".
- Bar stools: 30".
- Pendant lights should be hang 40-43" above the kitchen island.
- Dining tables: 30".
- Dining chairs: 17-19".
- Bathroom vanity: 34-35".
- Standard toilet: 16-17" height. Comfort height toilets for tall people or people with disabilities: 17-19" height.
Do you know Joe, Jay and Jim? They are renting a 3-bedroom apartment in Cambridge and have to split the $3,000 monthly rent.
But the trouble is that all 3 bedrooms are of a different size.
Divide the rent based on a room size?
Good idea! However one room has two windows, second has only one and the third one has a fire escape in front of its window. Joe really wants the last one for his cigarette breaks but he is wiling to pay no more than $900 for it.
Apparently math could help them!
Recent New York times article describes an algorithm that could be used in such situations. Take an equal-sided triangle where every vertex represents a totally unfair situation when one person pays for one room only the whole $3,000 rent. Subdivide this triangle into smaller ones and interpolate rent linearly into every new vertex. See below:
Now go over all the vertices and for each vertex ask either one of the roommates which room he prefers at such rent split. Apparently there always will be a triangle where every roommate has picked a different room. The fair rent subdivision lies somewhere in this triangle. Read more about the algorithm in the New York Times article: To divide the rent, start with a triangle.
And try the New York Times interactive rent division calculator: Rent Division Calculator.
Top image from Rentcafe.
It all went like in the kids' classic: "If you give a mouse a cookie." Stair redesign led to flooring changes and required a new kitchen layout; while we were at it we extended the kitchen into the garage that demanded new garage doors; since the floors were open we were advised to renew the pipes; then came the bathrooms and the doors. It all got so expensive that we gave up on the roof suite dreams and re-planned for it in the basement. At the end we thought that perhaps the original stairs were OK but with 5 workers and a cement mixer inside the house it was too late:).
Double the time and double the budget but still married, we are now only in the middle of the process. My book dreams were put aside, as well as summer travel. Museum and park visits got replaced with tile, parquet and furniture shopping. The whole adventure did give us a lot of valuable construction knowledge that unfortunately will not come to use as we swore to never ever renovate again.
As I was immersing in, more and more renovation math shoved on me daily by general contractor and designer. I was trying to keep our kids involved and stimulated by occasionally translating to them these real life math puzzles:
- If it took 3 days to destroy all the walls, how long may it take to build new ones? 1) 3 days. 2) one week 3) three months. #3 is the correct answer.
- Which room is bigger? Calculate the area and compare.
- Can you help me figure out how much tile will we need for your room? What about the whole basement (split into rectangles and calculate)?
- What is the length of the railings that we will need to order for the roof?
- How many stairs are we going to have? Our old stairs were surprisingly cumbersome: uneven, too high, slightly tilted. To lower the stair height we had to increase the overall staircase area extending it into the kitchen. The new stairs were supposed to be 7" each so my daughter and I calculated the number of these new stairs: (ceiling height + ceiling-to-second-floor thickness) /7
- How many 6" tiles could you fit into a 60" x 60" area? 100? We were wrong! There is a 3mm-wide grout in-between every tile connection.
- How many parts a bathtub shower and faucet may possibly have? More than you could see and imagine!
- How many white colors do you think exist in this world? One? Wrong again! But they all would look the same to you if it were not for your designer biasing your perception.
A few months ago, I wrote a post on how math could help us get more out of the home space we have. Now, looking around my home, I suddenly realize that without conscious intention, I put many of my advices to work. Observing the results, I believe that we now use our not-so-large home more effectively and enjoy it more than before. Here are the 6 simple things that I tackled:
My husband has size 14 shoes. They are almost 12" or 30cm in length each. It took me years to realize that he needs much more shoe closet space than I or any of the kids. But even when we divide it 1-to-3, with him taking a whole half of the closet and me sharing the other half with the kids, his shoes outflow his space. I do have hard time believing him that he does need separate sneakers for running, basketball, squash and hiking. But it is too late - all this (and much more) is already in the closet. One simple approach would be to divide the closet floor proportionally to the shoe sole area. But a truly ingenious geometrical solution is to consider the other dimensions. On this Ikea rack, small or large size shoes take the same space when arranged vertically.
Writing about the statistics of our use of each of the rooms, I created a home usability spreadsheet that can be downloaded here. Studying it, made me realize that our living room was vastly underused. We were barely spending 6 hours per week in it, that is 2.5% of our total home-spent awake time. Why didn't we use the new reading couches and bookshelves we recently purchased? Mainly because the light in the room was never strong enough for the evening reading. Last month we installed four cute little spotlights that are strong enough for late night winter reading or a family game. We put a few games, books and magazines on the table and encourage kids to cuddle with a soft warm blanket on the sofas to read, chat or play.
A discussion with my friend who recently moved into a new home and from all the rooms that needed renovation, decided to concentrate time and money first of all on the kitchen because "we all spend so much time there" made me review my family's room usability chart. Indeed, after the office (I work from home), our open kitchen-dining-homework area is the busiest in the house and perhaps it is about time we invest to redo the antiquated 70th style kitchen to something more fitting our taste.
Feel free to upload and modify this room usability spreadsheet to investigate what is being under and over used in your home.
Learning from a flight attendant's suitcase packing tips, I started rolling my towels instead of folding and immediately gained extra space in our closet. Plus, kids find it much more fun to help me now. However, their favorite rolling place is the floor.
Ever since we moved to the East Coast from LA, we were wondering why toilets and vanities seem to be much lower here. The standard 30" height of the bathroom cabinets apparently came from the old days when water bowls were just put atop a dresser. Such a low height may work well for kids of certain age and height, but little kids still need a stool and teenagers are the height of their parents. Therefore, unless you are running a home-school, you can be much more comfortable with a higher vanity that does not require so much bending. Especially, if you have a bad back. To test this, we tried hand washing and teeth brushing in a kitchen sink, as kitchen counter height is usually 36". We liked it and when redoing our bathroom after an occidental flooding, we knew that we wanted to fit our home to our "tall" needs.
When turning our utilities room into my husbands' office and contemplating the style of the bi-fold doors that should cover the washing and drying machines, I recollected my own advice: "Mirrors make room appear larger and brighter by reflecting the light instead of absorbing it as walls do." Now, we can't believe that we got so much visible space out of the junkyard of utilities and storage.
Please feel free to post in the comments your own advices and stories on creating more space or making home more efficient with the use of trivial and not so trivial mathematical thinking. You can also share your own Home Usability Chart with your kids discussing how to make heavily used rooms more fun and underused rooms more practical.
When our kids were little, hardly a day went by without someone spilling a glass of milk, juice or water. We were prepared – with sippy-cups, plastic cups, paper cups and abundance of paper towels. We knew that while the kids’ motor skills are still developing, ours are too sleepy to function as designed, and the kitchen is too small to allow the juggling circus we were trying to perform there. Still, there were days when I thought there is a spilling spell on our family. Then, I started blaming the cups. Plastic or paper – they were unbreakable but so light and unstable that just a light push would flip them out of balance.
Ever since, I have been preoccupied with the search for a perfect drinking cup. Take a simple, clear water glass. Every houseware store has a wide selection, every season brings new designs. But have you found the one that works for you? What qualities does it possess?
Learning from the kids’ plastic cups lesson, I defined my first requirement as stability on impact – the perfect glass should be difficult to take out of balance. This means it needs to have heavy and/or large bottom and/or thick walls. The retro-style glass pictured below was proudly purchased by my husband, and it seemed to satisfy my first demand. Was is perfect? Nope - too heavy to lift even for adults, and too bulky to store. Plus, while it may work well in a diner, I did’t like the look of it on my dining table.
So, stability was clearly not enough, and the list of properties of the perfect water glass started growing:
- not heavy
I bought more and more: Ikea, Crate and Barrel, Cost Plus, Pier Import, Marshalls. Some were too short to hold the amount of water accompanying a meal. Some were too tall to fit in my kitchen shelves. Some broke on the first week, some I couldn't break as much as I tried to. A few years ago I thought I finally found it. The perfect glass: stable, light and elegant. The dips on the sides made for a sure and easy grip. I bought 12 only to realize that I can’t stack them, plus the cave-like space inside the glass doesn’t allow it to be cleaned well in the dishwasher.
Back to the list. The perfect glass should be:
- not heavy
- stackable and not too tall for efficient storage
- large enough to hold 1 cup of water
- cleaned well in the dishwasher
I read about a design exhibition in Copper Hewitt Museum in NYC that showcased elegant Swedish drinking glasses “which swell gently outward just below midpoint.” The swell was a special design feature to stabilize the grip so that the glasses could be comfortably held by people afflicted with neural or muscular disorders that produce numbness or tremors.
These glasses would look nice on my dining table, they appear stable and easy to hold even for a child. But they seem too gentle for a daily hectic kitchen life.
I decided to go back to the simple solution and finally settled on something that looked OK, was moderately heavy, rather sturdy and looked promising to become the one and only.
At the store I went over my list:
- stable – I sneaked a light push that didn’t affect the glass
- not too heavy - yes
- beautiful - OK
- stackable and not too tall for efficient storage - yes
- large enough to hold 1 cup of water - yes
- cleaned well in a dishwasher – should be because it is smooth
- not too gentle for a daily handling – thick enough yet not heavy
But guess what, some sandy grains are always accumulating at the bottom of the glass in our dishwasher. I suspect it may be too tall and narrow to be cleaned well automatically.
I am still searching for the one and only. But does it exist? Have you found one?
It used to be that a small talk would start with the topic of weather. This month, however, it is: “How is your refinancing going?” You will bond in an instance and also find what today’s rates are. Like a circus animal that is eyeing a treat it could snatch after hard work, I motivate myself to think about refinancing by visualizing all the wonderful things I could potentially do with the money saved by it. $100 per month – an extra shopping trip, or let it accumulate to a $1,200 per year – you got a small family vacation. Wait a few years and use this for a new kitchen. Definitely worth investing a few hours to play a math game with your lender and then copy and sign the pile of paperwork.
Here are the rules of the math game:
- you prefer to pay less than you already do on a monthly basis
- you do not want to increase the overall sum of your payments from now till the end of your mortgage or sale of the house
- with every year you own more and more of your home, you do not want to significantly slow this process (named equity buildup), and in fact you probably prefer to speed it up
With the rates falling so low throughout the last year, we all probably got some sense to obtain fixed mortgages. If you plan on staying in your house for longer than the fixed part of your adjustable mortgage is, you may want to grab a fixed one.
So, you have likely refinanced in the last year to a fixed rate or just a better rate mortgage. Now, rates fell again. Should you refinance? Refinance now or wait? This has been the main topic of discussions in my neighborhood, while we walked the dogs, compared whose grass is greener, or watched the kids play outside. We read some articles; we talked to our mortgage lenders; we bounced around some numbers. Here are the refinancing basics and some tricks we learned along the way. Neither of this should be taken as a direct advice but rather some food for thought.
What you would need:
- Call your lender to find out your current rate, how much you borrowed, how much you currently owe, and how many years you still need to pay.
For example, say you borrowed a $417,000, for 30 years, with a fixed rate of 4.875%
1.5 years later you still owe $407,000.
- If you think you may owe more to your bank than the current price of your home – beware of appraisal, the bank may not allow the refinancing. Check this before going any further.
- Ask your lender or shop around for the best options for your refinancing.
For a $407,000 amount the following options may be considered this week:
Option 1: 4.25%, 30 year fixed with closing costs of $2,600
Option 2: 4.375%, 30 year fixed, no closing costs
Option 3: 4.375%, 25 year fixed
Option 4: 4.25%, 20 year fixed
- Start comparing your current mortgage and each of the alternative options. See my colorful table below. I used Bloomberg mortgage calculator to fill in details for each of the mortgage options.
All the dollar amounts marked blue in reality are smaller than they are. Part of them is tax-deductible. See more details below.
Refinancing costs unfortunately are NOT tax deductable.
The key question in refinancing is how long you think you are likely to stay in the house. Rows 6, 7 and 8 describe a few possible scenarios. In many cases, one refinancing option is better in the short term but another on the long term.
The numbers in lines 6, 7, 8 are obtained by pressing View Report button on the Bloomberg mortgage calculator. Look at the table in this report. What you owe is your Ending Principal Balance, last column of the report.
What you paid is: (monthly payments) x 12 x (number of years).
Number in line 9 is your total payments. It is listed at the top of the report as well as inside the Bloomberg’s calculator GUI, below the monthly payments.
Now, let the math game begin:
Shall you consider refinancing? What option to choose
- In general, the more you owe and the longer you still need to pay, the more it makes sense to refinance. Smaller interest rates shrink your interest. And in the first years of your mortgage term, 70-50% of your payments are part of the interest. Once you get to the tail of your mortgage, you are mostly paying principal and shrinking the small interest fraction by extending your payout time doesn’t make sense.
- Shorten your term:
If you still have a large chunk of your mortgage to pay, you are very likely to find a profitable refinancing option in the current environment. Compare each of the candy-shop colored columns with the yellow one (your current mortgage). Note row #4 (monthly payments amounts), row#5 (number of years to pay) and #9 (total accumulated payments). If you are doing well financially right now and can afford paying a bit more monthly, you can get advantage of the low rates and shorten your mortgage term. In our table these are last Pink and Grey options. Look how significantly it saves us in the example on a long term: $150,000 before tax! Monthly payments are higher but many people get part of them back in taxes. The actual payment difference is: (Payment difference) x (1 – tax rate)
In the example, if we assume 28% tax bracket, the actual payment difference between yellow and grey options it is:
($2,520 - $2,206) x (1 – 0.28) = $314 x 0.72 = $226.08
- Save monthly and in the accumulated payments:
If you are struggling financially, you can use advantage of the low interest rates and refinance with the same or slightly longer term, reducing your monthly payments. In many cases, it will reduce your total accumulated payments as well. In the example, these are options green and orange, 30 year fixed rate with and without closing costs.
- No closing costs vs. closing costs options:
The no closing cost option is almost always your best bet if you are not certain you would stay in the house for the whole term of your mortgage. Interest payments are tax deductible. Closing costs are not. Closing costs are like a Costco membership fee – worth paying only if you know you will get the money back in savings throughout the term of use. What if you need to move? Life is unpredictable: job relocations, family dynamics, kids’ needs or new hobbies. You loose if you move soon after the refinance. How soon is this soon? Calculate how many years will it take for you to recover these up-front closing by comparing closing costs option to a no closing costs:
Months to recover = Closing costs / monthly savings
Monthly savings are = (Difference between mortgage payments) x (1 – tax rate)
In the example, the difference between green and orange mortgage payments is $30.
Monthly savings are = $30 x (1- 0.28) = $21.6
Months to recover from the upfront $2,600 closing fee =
$2,600 / $21.6 = 120.37 months (that is over 10 years)
Only after these years you will start seeing the benefit of closing cost as compared with the no closing cost option. So, consider this option only if you certain that you will stay in your house for at least as long and if you believe interest rates won’t be falling anymore tempting you to refinance again.
- Wrapped closing costs trick:
If you really like your closing cost option but don’t have any open cash to pay for the closing costs, check this interesting trick: instead of paying closing fee up-front, borrow more equity and use these extra for closing costs. This Option is described in the blue column of our colorful table. If you are still able to reduce your monthly payments as well as overall accumulated payment and plan to stay long in the house – consider it. Why stay long? Because if you sell or refinance in a year or few – you will still owe to your bank these extra closing costs you wrapped into your mortgage. Therefore, this option is quite an opposite of pink and grey – instead of speeding up, you slow your equity buildup on your home.
- Invest then pay off:
With the interest rates so low and interests being tax deductible, you are paying very small rates on your loan. For example, on a 4.25% loan with a 28% tax rate, you would pay only 4.25% x (1 -0.28) = 3.3% interest. So, for every $1,000 bank offers you today, you would have to return only $1,033 in a year. There are those of us who are confident that they could generate much higher profits in a stock market or other investments. For them, an extra $1,000 today with a 10% profit and 20% tax rate on a long-term investment will result in an 8% profit or $1,080 in a year. So, instead of using any extra money they may have to pre-pay or shorten the mortgage term, they can profit more by investing: $1,080 - $1,033 = $47
Do your math, consult with your lender, chat with your neighbors, and pick what works best for you. All the above is not a specific advice but only points to consider. Interest rates do deviate a lot, so save your comparison tables for the future. Whether you refinance or not, remember: you just cleverly saved by doing something or deciding not to. And as soon as you start receiving those monthly savings, celebrate and treat yourself to something memorable.
Acknowledgments: I appreciate the help of our lender, Connor Shortsleeve (leaderbank.com) and my neighbors in preparation for this article. Top image from Flckr, distributed under the Creative Commons License
Ready to follow your expansion dreams? Here is your guide on what you can do. The guidelines vary from town to town, and could be adjusted at any time, but you need to know the basics in order to adapt to these changes and know what you can ask for. Sit together with an architect, and play with some math to see what space you need, how much you could spend, and whether your goal is to increase the official home living area. Then, you might consider consulting with an experienced real estate agent or appraiser to see what your home’s increased value would be after the expansion.
In order to check your expansion options, you need to start by figuring out your current home size. It’s like the starting point of a dieting journey, except that here we hope to gain rather than lose. How do you measure it? Pretty intuitively: take your home plan if you happen to have it (such as the one displayed below) or create one by drawing a bird's eye view of your home. Go around the wall perimeter, measuring and marking every wall. For homes you usually measure around exterior walls, for condos – around the interior. Adjust the shape if necessary, then split it into rectangles, compute the area of every rectangle (length of one side times the length of the other), add it all up, and voilà. But, wait! Do you count the bathrooms? What about the staircase? And the back porch that you just turned into a sunroom? Is it counted as part of your square footage?
These mystical GLA initials next to the square footage in your mortgage papers or appraisal report, as it turns out, mean Gross Living Area. The GLA is defined as the total area of finished, above-ground, residential space, excluding unheated areas such as porches and balconies. See the sample home (Dover design) and the floor plans from the CadSmith Studio LLC
All the areas highlighted in pink and blue are counted as part of the GLA.
1st floor: 28' x 38' + 16' x 24' = 1064 sq ft + 384 sq ft = 1448 sq ft
2nd floor: 28' x 38' = 1064 sq ft
Total GLA: 1448 sq ft + 1064 sq ft = 2512 sq ft
Here are home areas in question that are counted as part of the GLA:
- bathrooms are in
- halls are counted
- closets are in
- even the dead space inside stud walls is calculated as living space (except for the eaves in cape style and some other styles of homes)
- staircase area is counted twice – once for each floor
- porch turned into a sunroom is counted if heated and used year around
- high-ceiling finished and heated attic is counted
Here is a list of areas NOT counted in the GLA:
- open foyer -- essentially a two-story hole that is not counted twice, but just once
- porches and balconies (unless heated and used year around)
- finished basement – surprise, surprise - is out of the GLA unless it is completely above ground level and used year round; however, an appraiser will usually include them in a separate category of “basements and finished areas below grade.”
- low-ceiling attic (see details below)
Ok, we know what we've got now, the current GLA. Ready to expand?
First, beware: increasing your GLA will likely lead to greater home value but also will increase your taxes.
According to broker and appraiser Sam Schneiderman, President of Greater Boston Home Team, the least expensive way to create an extra room or two of living space is usually to finish your attic.
Will a finished attic increase your GLA? Yes, if you can reach it from inside your home, without sneaking through the garage or romantically ascending exterior stairs, and if it fits your town's legal room height requirements. In some neighborhoods, for instance, the portion of the attic that has at least 5' height and an average ceiling of 7' is counted as legal GLA. Check your numbers carefully with your local building inspector and appraiser, keep it legal, especially if you plan on placing your mother-in-law or live-in nanny there.
If your attic is too small or non-existent, consider insulating and heating your garage or basement. Remember that the basement most probably won't add to the official GLA numbers, unless it is completely above ground. But then do you call it a basement? :)
If you do decide to expand, then you have to know a bit of expansion math. Luckily, it is quite logical. Unluckily, each town has its own rules and may have their own methodology when it comes to calculations. The most common rules according to Mr. Schneiderman are:
Setback Rules: To keep peace with neighbors, guard privacy, and avoid fire hazards, town building codes require some minimum space between your house and your neighbor’s house. They are called setbacks and vary at the front, side and back of your lot. Setback space is off limits for construction. You can find setback requirements in your town's building code. Then, take your lot plan and mark these “buffer” areas off-limits for any new construction. (If you can only expand by encroaching into the setback areas, ask the town building inspector about the likelihood of applying for and receiving a variance.)
Max Lot Coverage Rules: Still have some space left on the lot plan to expand sideways or up? Check your Max Lot Coverage % in your zoning district. There could be a limit on how much of your lot can be covered by your house. If applicable, maximum coverage rules are calculated as follows:
House Footprint / Lot Size <= Max Lot Coverage Therefore, you can expand as long as: (House Footprint + addition) / Lot Size <= Max Lot Coverage Your house footprint is just what it sounds like – square footage of the print your house makes on the lot.
Floor Area Ratio (FAR) rules: Most towns usually limit how much living area is allowed on lot of a given size. The limits can vary depending on the neighborhood or zoning district. Check the Floor Area Ratio (FAR) to see if you can expand your GLA. If applicable, the FAR rule is typically:
GLA / Lot Size = FAR
Therefore, as long as:
(current GLA + addition) / Lot Size <= Maximum FAR you can expand without a variance
Maximum Building Height Rules: As you may have suspected, you can't build an Empire State Building on your lot (because town fire equipment may not have Superman on staff to reach the top if it catches fire!) Most areas have a maximum height limit on buildings. Again, your town's building code is your bible. In my neighborhood it is 2.5 stories and 35'.
So, learn your expansion math dance with your architect. Ask your town's building inspector to approve your steps (calculations). Only after that, hire your contractor partner and get started with measured and riveting twirls. It all requires team work. Don't skip your steps.
While it may seem like a lot to consider, the math of it all is elementary. Beware that rules vary from town to town, and are updated from time to time. Your bank appraiser's calculations may disagree with the town building inspector's, but now you know what they are talking about and what you should be asking for. Enjoy playing with these numbers to plan, dream and negotiate your home's expansion to meet your needs and realize your dreams.
Meredith and Derek dream up the plan for their house on Grey's Anatomy
P.S. I would like to thank Sam Schneiderman of Greater Boston Home Team, Bonny Smith of CadSmith Studio LLC, Needham (MA) building inspectors as well as Anat Yosfan of AnatYDesign for their help in preparing this article.
In the following two columns, let's explore how math can help you better utilize the space you already have, and if desired, also guide your expansion plans.
Part 1: Make the Best of the Space You Have
Whether you occupy 500 square feet or 5,000 square feet, you are likely wishing you had more space for storage, work or entertaining. The best home advice for those running out of space: instead of figuring out what you can add to make things work, think of what you can re-arrange to make things work better. By merely shuffling your furniture and decorations around your house, you can create the refreshed and practical environment that you are looking for.
Think statistics: is your living room empty 25 out of 30 days a month, while you work out of a stuffed tiny home office? Of course you have hopes to populate this living room with book reading or chess playing kids on a daily basis, even if past experience shows they are more likely to be found in the family room watching TV or doing homework in the kitchen. Don't wait for your life style to fit your home, make your home fit your needs today!
To get a clear picture of these needs you may want to create an occupancy chart of your home. List all of your rooms and family members, and then add up the hours that each person spends in each room every day. Analyze the results with your family. Use this 95% idle living room to create a glamorous CEO office for yourself, make dreams come true with a pool or sewing table, or expand the busy dining/homework area.
Once you start thinking about how to use your space more efficiently and fairly, it is guaranteed to get addictive. Let's do some basic geometry. If your shoe size is 10 and up, the total square footage covered by a pair of your shoes is at least two times the square footage covered by the shoes of your 1st grader. Feel free to use this argument to snap up at least twice the space in the family shoe closet.
Not everyone is created equal. If your height is above average, you may have noticed that you need to bend 90 degrees over most bathroom sinks. It is, however, a nice stretching exercise, after sitting on that dwarf-ish toilet seat (14” height is the most common). But how can you check that your hands are clean when you bend so low over the sink you are in danger of summersaulting into your mirror image? On the East Coast, such low toilet seats and vanity varieties are surprisingly very popular, either due to a low average height of the original settlers, or because we put our kids' needs way ahead of our own. Consider treating yourself to height-appropriate bathroom furniture during the next round of renovations.
Have you always assumed that your floors are straight until your kids showed their toy cars sliding in one direction? Have you ever hung a picture using a level and then discovered that it is tilted with respect to the walls? This may be a hint that your floors are not level and/or your walls are not perfectly vertical. Not to worry: align the picture with the closest wall, forgoing level and gravity. Similarly, use pads to realign your furniture. And feel free to blame such floor tilts for the natural accumulation of body fat midway up front or at your bottom – its your body's attempt to balance your center of gravity.
Are you looking for creative ideas to materialize more storage space? Zoom in on your usability chart. Look for some idle corners or niches. Our ingenuity sparks when facing a real challenge. Do you happen to have both, shower and bathtub, in your bathroom? An unused bathtub and everything above it could be turned into a nice size closet of around 5' x 2.5' x 7' that could hold your linens, towels and travel luggage filled with winter sweaters.
Be generous with mirrors as they make rooms appear brighter by reflecting the light instead of absorbing it as walls do. They can fool our perception of space, making any room look larger. But remember the laws of reflection – if you can see something, this something can see you – mirrors can limit your privacy. To increase space not only perceptually but also physically – you'll have to learn the intricate steps of expansion math dance with your architect, town building inspector and your contractor. But this is another story - for next week.
To read more about mirror reflections click here.
So, how does one fold and pack optimally? Pack so that all the items stay neat. Pack so that it takes the least amount of space. Pack so that one could easily find any single item and take it out without a major disturbance to the other items.
Packing problems extend well beyond clothes – just think of all the large boxes you've received, ordered through online commerce, that contained small volume items and lots of Styrofoam peanuts or bubble wrap or air bags. Dissertations have been written about optimal packing, scientific wars fought and large prizes awarded. People have been inventing square tomatoes, space bags, nested dolls, and new polymer materials - just to minimize the packing headache.
Who would really know how to pack clothes optimally? Of course business people and flight attendants who manage with one carry-on bag for a week. In a slide show for the NYTimes Heather Poole shows that it is not only possible but also easy: 10 Days in a Carry-On
She rolls her clothes instead of folding them! What an ingenious idea. It does work well for sleeping bags, why didn't we think of using it for clothes? Tight rolling pushes the air out of the clothes, they wrinkle less, and it is easier to take out one piece without messing up the others. I decided to check how this would work in my towel closet and recruited my daughter to help :
Our twenty towels folded:
Same twenty towels rolled up:
It looks like we have gotten around 15-20% of empty space in the rolled version. It is also much easier to see each towel. My daughter could easily pull her favorite one from underneath without turning the whole closet into a battlefield. In a folded scenario some tiny towels are obscured – I could spot only eighteen out of the twenty. Combining small and large towels on one shelf is much easier when they are rolled. No collapsing towers of folded towels of various sizes. And most importantly, my daughter and I agreed, rolling was much more fun! One small change in our habits - one large optimization step for our closet.
So, let’s Roll n’ Rock.
Do you have your own packing and folding tips to share? Please post them in the comments below.
My husband, on his return from a business trip, puzzled me when he remarked that the “purple” carpets seem too dark. “They were very brown when I bought them,” I said and rushed to double-check. After spending two frustrating months on choosing our home wall colors, I knew very well his strong dislike of anything purplish. Surprisingly, the carpets did look dark: purple or brown, one could not say. My deep purple nails appeared to be identical to the carpets.
So, I went on a quest, trying to figure out how two very different colors, brown and purple, can appear exactly the same. And when does purple become brown?
Playing with an online Color Tool, that decomposes any color into Red, Green and Blue components, I realized that both, brown and purple, have very little green in them, and much more red and blue. By taking brown and adding some blue to it we get purple.
So, the answer to the mystery should be in blue. Aha! Perhaps we cannot distinguish between brown and purple in some lighting? Turns out, this is true! “White lights” such as strong skylights and white fluorescent lights as in shopping centers have a lot of blue in them. See the hill on the left graph above the blue. “Yellow lights” such as sunlight and the incandescent light of my bathroom have much more yellow and red than blue in them – see the curve on the right picture.
The color of the object that we perceive is a combination of its color and surrounding illumination (light coming from the light source such as lamp, sun or reflected from other objects).
A simplified color equation:
Object color that we perceive = Object’s “true” emitted color X Illumination
Therefore, any blue in the object is amplified by a skylight or fluorescent light, making it appear even stronger blue to our eyes. The yellow light of the incandescent bulb or direct sunlight on the other hand, highlights the warmer tones, suppressing the blues. That explains why my brown carpet and my purple nails appear exactly the same in the yellow-ish bathroom light and look very different in the fluorescent light of the store.
What is the use of it all, you may say. It tells us that instead of exclusively considering the color of our purchases at the store, we should consider the color as it appears in the lighting conditions this purchase is intended to impress. Buy, try, and perhaps return. Otherwise, your Holiday purple dress may look too brownish for your taste, and those shiny light purple eye shadows may not be a good match at all.
If you are carefully choosing a fancy blue shade of color for your room – make sure you have plenty of white light in it, otherwise your visitors will just call it plain blue.
And if your kid comes home with the wrong color jacket, don’t worry about color blindness and don’t argue – everything will look different in the morning light.
Turning the spending awareness on and imagining myself as a financially savvy Desperate Housewife, I realized the advantages of a liquid soap refills. First advantage is minimizing the vicious cycle of plastic manufacturing and recycling that we are paying for. Those small liquid soap bottles that were produced, packed, shipped, stored and sold, you use them for a month or so and then recycle, with many more vehicles driving it from your home to the dump, to the recycling facility, washing, melting and starting their life cycle all over again only to return to your home a few months later. Sounds like too much hassle for such a short life-span of a little bottle. And obviously, we are paying for every step of this process.
The second advantage is your direct saving with a surprisingly minimal amount of work. A 500 mL 16.9 oz plastic bottle of liquid soap that I bought in Marshall’s was $5, with its original price being $9. Costco is selling small size bottles of soft soap containing 7.5 oz for $2.23 as well as a gigantic size 1 Gallon (128 oz) bottles for $13.99. As usual, to compare we need to check the price for the same amounts - say 1oz. The liquid soap from Marshall’s as well as a small packaged soap from Costco were both $.30 per 1 oz. Surprise! Wholesale prices are not always cheaper. However, the gigantic soap from Costco was $0.10 per oz – 1/3 of the price I am currently paying. We could save 2/3 of the price and simplify our life by buying the soap only twice a year instead of monthly! So, consider refills. They are easy, green and hip nowadays, especially among desperate housewives.
Look for TheMathMom's article in September 2009 issue of the Boston Parents Paper on many creative ways math could come to the rescue of your family budget.
Click here to read this article online.
Consider these three potential mowing patterns.
Which one should you pick and why? Obviously, you want to walk less. More importantly, you want to consume less gas and make fewer sharp turns, as it is rather inconvenient with a bulky mower.
Assume your lawn is 50' by 100', and the blade of your mower is 2' wide.
In the mowing pattern of vertical parallel lines (left), you will walk 25 stripes of 100' each, 2500' total. You will turn half a circle (180 degrees) 24 times.
In the second scenario, horizontal lines (center), you will walk 50 stripes of 50' each, 2500' total. You will turn half a circle 49 times, twice more than in the first case! (You should try turning 180 degrees with a heavy mower to understand how un-fun this is going to be.)
In the third case (right), you will be moving in concentric motions, and you will walk:
100' + 48' + 98' + 46' + 96' + ... + 56' + 4' + 54' + 2' + 52 '=
(100' + 98' + ... + 56' + 54' + 52') + (48' + 46' + 4' + 2') =
152' x 12.5 + 50' x 12' =
1900' + 600' =
Same distance! But you will make 48 90-degree turns, which are much friendlier than 180-degree turns.
Overall, the concentric pattern is your best choice. It does not save you any distance, but it allows you to avoid pirouetting with your mower. Only sharp parade turns. Better yet, forget about lawn corners, smooth your mowing angles and turn concentric rectangles into elegant ovals. No sharp turns at all! Just like Google's goats.
Read more stories about math of cooking, dating, parenting, home management and travel from The Math Mom: www.TheMathMom.com
Planning any air travel? See why airplane flight route looks suspiciously curved while the shortest distance between two points should be a straight line.
It is intended to look correct in the rear view mirror when the ambulance is driving behind us, but looking at it straight on is disorienting. Someone, please come up with a creative, non-confusing solution that works both ways!
The masterpiece of reflection is a flat mirror. Every ray of light is elegantly bouncing back at exactly the same angle it came in, creating a virtual image identical to the one in front of a mirror. Or is it? Doesn't the image in a mirror look like it is a right-left flip of the original figure? Don't we see ourselves with our right hand being left and left hand being right in a mirror? And if this is true, why do mirrors only flip right-left and do not transform up-down?
I tend to agree with those who believe that mirrors have never pretended to do a right-left transformation. With its parallel reflection, a mirror is flipping our image in the depth dimension (Z axis). Your right hand stays on the right and left hand on the left, top on top and bottom at the bottom. We are basically looking at a Xeroxed copy of ourselves.
But our brain is unsure. Much more often than seeing ourselves in a mirror, we are used to seeing other people in front of us. Standing, sitting or pictured with their faces toward ours, with their right hand in front of our left hand and their left in front of our right. When comparing such a figure with our own mirror image, it does look right-left flipped.
Right-left flip comes out of intuitive comparison to something we are accustomed to seeing. But why should we be comparing? If we constantly saw gymnasts on the bar rotated upside down, we might want to compare them to our mirror image and it then would look up-down flipped.
So, let's stop blaming the innocent mirror for confusing us. It does only one thing, and it does it very well: object reflection in depth, aka Xeroxing. It has never pretended to do left-right or up-down flip. The flip is our brain's desperate attempt to make sense of a virtual object we see, by matching it with something we know well.
And as for visual confusion with the mirrors, it is my excuse for skipping mirrored aerobics rooms. There are too many lefts and rights there.
Like it? Try more math adventures in home ownership by The Math Mom: Pythagoras vs my husband... or You put a dirty plate in, you take a clean plate out. And you repeat it all around.
Recently, my husband was finishing the assembly of another one of such exemplars – a tall bookcase - on the floor of his home-office. He realized that he could not rotate it to standing position because the ceiling was too low. For a couple of minutes we were very quiet, feeling a bit stupid.
We did select this particular bookcase because its height was just inches shorter than the height of our ceiling, proud to use our space so efficiently. But we did not foresee that the bookcase's front diagonal would be larger than the room height (85” > 82”), preventing the installation... Humiliated and frustrated, we were waiting for that spark of creativity that occasionally saves you from doing things all over again. And then I saw this triumphant smile on my husband's face. A minute later the bookcase was standing up tall...
The side diagonal obtained from the height-depth triangle was shorter than the frontal diagonal obtained from the height-width triangle and smaller than the room height (76” < 82").
Next one on our shopping list is a grand piano. I think I will start from Craigslist. What can possibly go wrong?
Read more from The Math Mom about fun and hip math of our routine existence: The Eureka Moments Of My Household.
Confused by the imminent digital TV switch deadline, and lured in by advertisers promising something bigger, flatter, brighter and sharper, many of us went and bought new HD TV sets. The high-definition picture does appear better because it is digital, has a greater resolution, and you usually watch it on a larger screen. But the HD format also has a very different proportions (width-to-height ratio, called aspect ratio) of 16:9. It is the same as a movie theater screen. The problem is that most of the TV programs are still not in high definition format. They are produced in a standard format with 4:3 width-to-height ratio. Ratio of 4:3 is the same as 16:12 and 12:9. This means that when we fit a standard TV program or a movie (with proportions 4:3 or 16:12 or 12:9) into our new HDTV screen (with proportions of 16:9) we have to either stretch the picture, or zoom and cut some of the frame.
Let's take Patrick Dempsey, voted TV's hottest gentleman. Assume that Grey's Anatomy is broadcast via a standard definition TV signal with image proportions of 4:3.
Displayed on a high definition TV set, the standard definition image is lost against a black background:
HD TV sets allow us to enjoy the wide format of the screen by offering a number of display options for a standard definition signal. The most popular among them are zoom and stretch. The Zoom option leaves the image undistorted but cuts the image’s margins, which frequently includes people's heads:
Remembering how much this TV costs and dissatisfied with missing information from the frame’s margins, many choose the Even Stretching or Full option. It distorts the content, but nothing is lost:
When image of standard TV proportions of 4:3=12:9 is being stretched horizontally to fit 16:9 aspect ratio, what has been displayed on 12 inches becomes 16 inches. Everything starts to look 4 inches wider, which can be translated to 1/3 fatter. Surprisingly, we are getting used to it quickly. And who knows, maybe it can explain obesity...
The scary part about this is that for many of us, TV is a window to the outside world. We frequently believe things are as they appear on TV. As we get used to newsmen and women with round faces, and compact cars stretched like limos, we may be disappointed to discover the real size of things.
If you are a big fan of Grey's Anatomy you will like this story from The Math Mom: The Math of Ex.
But what if, in addition to being ungrateful, I am actually the one to blame for the non-intuitive or inconvenient grouping of my kitchen shelving items? Are dessert forks more similar to dessert spoons than to the main meal forks? Shall we group by shape similarity or by function? How can you optimize your kitchen so that anyone can find anything and minimal effort will be spent on miscellaneous organizational tasks such as finding serving utensils, cleaning the table, and emptying and filling the dishwasher?
Interestingly, it took a working couple with a dozen kids to come up with a way of making this kind of efficiency a science. A time and motion study was devoted to reducing the number of motions in a task in order to increase its productivity. Frank and Lillian Gilbreth even creatively named a unit of this effort as a “therblig” - inversion of their last name with 'th” as one letter. By observing a bricklayer's job, they reduced the number of motions for laying one brick from 15 to 8, increasing productivity and decreasing fatigue – just what we need in the kitchen.
A friend of mine who made her career in being organized from the start has actually organized her kitchen with minimal therbligs in mind: sugar and teaspoons are right next to the coffee machine; scissors, stamps and markers await mail near the entrance. All unneeded movements are eliminated. She also labeled every single bin of the dishwasher and all the house shelves with the name of the item that should be stored there. The house looks like it came directly from the movie “Memento” where the main character lost all his short term memory and had to write everything he was constantly relearning on his body and the walls of his room so he would know it when he needed it. Not everyone digs this approach. Owners have to withstand plenty of jokes from visitors and occasional mischief (guests marked my friend's shoes with right foot/left foot stickers). But things do stay in order in her house, and after helping myself to a cup of coffee I realized that an intuitive and efficient arrangement makes everyone feel at home.
Try it at home:
See if you can organize your house with minimal therbligs in mind. Fast and intuitive. For you and your guests. Can guests staying overnight find clean towels and make tea or coffee in the morning? How many steps does it take you to change wet sheet in the middle of the night? Are you shelving plastic dishes low enough for your kids to serve themselves? Encourage participation and simplify your life.
Read more stories by The Math Mom about hip math of home ownership: How Patrick Dempsey and HDTV may be responsible for obesity or Color Dilemma.
At the beginning of a new weather season, when the contents of your wardrobe need to be swapped, you sort clothes by colors or type or style, and neatly fold them on shelves and bins in your closet. If your family is similar to mine, in a matter of days you will observe that the closet has a life of its own. It has rejected your naive intentions of order: someone has placed your son's sports T-shirt into your long sleeve shirt bin, or mixed pajama bottoms and sports pants, or even rolled a dirty red sweater into a shadowy corner.
When would you officially call this a mess and order re shelving? How about utilizing mathematics to define a measure of order versus mess? All the literature says that children and husbands need very clear instructions. So, let's create closet clean-up guidelines for them. We can define a ratio of the number of correctly shelved items to the number of misplaced items. Signal-to-noise ratio (SNR) is the scientific term. Assume we have 50 articles in the closet. At the beginning of the season the SNR is very large: all items neatly ordered; nothing is misplaced. In two days we still have 47 ordered items and only 3 misplaced. SNR is 47/3 = 15.6. Doing well so far. Two weeks later you spend 30 minutes searching for your shirt and realize that half of the items have been misplaced. SNR is 25/25 = 1! Time to re-organize before SNR sinks even lower and your mess overpowers your order.
The original Signal-to-noise concept derives from the field of electrical engineering, where it is used to measure the strength of a signal compared with the strength of a background noise. For example, your phone has high SNR unless you get occasional interference from neighbors' calls or a static line noise. Informally, SNR started to be used as a ratio of useful information to false or irrelevant data. Homeland Security Information Center uses this concept to define how many of the daily received security alerts and leads should be pursued.
To me signal-to-noise sounds like a perfect way to measure the trustworthiness of friends or magazines. I love catching up with the latest gossip while standing in line at the store, but doubt that most magazines I glance at would get a SNR above 1. Wouldn't it be great to use SNR to define an official trustworthiness approval rating and stamp it on each printed or aired piece? Otherwise how on earth would we know if Madonna really had a fling with Yankee's Rodriguez?
Would you like to try something else from TheMathMom? See how math can help you test awareness level of your teenager remotely.
But wait, a top view plan of our house looks very much like a map. It means that four colors should be enough to paint all the rooms with no two adjacent rooms having the same color:
So, among four members of my household, we only need to choose four colors. One each. That's doable!
For more household adventures with math, try Cheap and Hip but Assembled by You
You see - math is hiding in every single room of our households, including even bathrooms and toilets. Why does the water go up when you are desperately trying to push the plunger? Although a plunger is as far from the golden wreath as any object could be, it similarly pushes away the water proportionally to its volume as Archimedes discovered during his Eureka moment in the bathtub. It is interesting that it does not really matter what is your plunger made of, metal or resin plunger would push out the same amount of water if plungers are of the same size and shape. But buying this new orange UFO-shaped plunger at the museum store was not necessarily a good idea. Mere immersion of it into my toilet has caused a serious flooding. Some things belong to the museum.
Now follow me to the basement where the washer and dryer are. This is where a little math has made one of the most strategic breakthroughs of my household. After a year of weekly laundry routine and the occasional mid-night crisis wash, we were looking for the bottleneck. The answer began with a simple question: “What is the first item that we run out of?” Not very sophisticated analytical investigation pinpointed that our laundry demands have been dictated by the lack of clean underwear. A trip to the store and $50 has allowed us to run half of the washing cycles per month. This has saved plenty of detergent, lowered our gas bills and most importantly allowed us to watch one more movie each week.
Try more stories about cool math of home ownership: How Patrick Dempsey and HDTV may be responsible for obesity or The House Hunting Story.
We had just a perfect wall for it. My dear husband, who rarely applies his perfectionism to household chores, proudly demonstrated to me on this occasion that he was using a level to ensure it hang straight. However, as soon as the last nail went into the wall, it became obvious the picture was skewed towards the nearby corner.
This contradiction dazzled us – shall we trust what we see or what we measure? Do I side with my husband and his brand new set of Home Depot tools? Can I just avert my eyes from this giant, misaligned 1970's piece of artwork in the middle of my living room, or shall I fight for the visual re-adjustment that would contradict the laws of gravity? Uncomfortable with either option, I continued to investigate and soon realized that an imperfection must be the answer to this puzzle. Either one of our house walls was not vertical, or the picture we hung was not a perfect rectangle. Both options were not very attractive, but at least they kept the peace in our household and maintained the laws of nature.
Now it was a matter of measuring the picture corners. This could be done with a simple rectangular sheet of paper or a measuring tape. Remember the Pythagorean Theorem? We frequently forget that its converse is also true, and in any triangle if its sides fit the equation x2 + y2 = z2 then the angle between sides x and y is 90 degrees.
To be honest, I am afraid to perform this measurement. What if our house wall is not straight? Imagine how many new mathematical problems this can generate.