A Wolf, a Sheep and a Cabbage - Your Daily Carpool Adventure

Remember the childhood riddle about a traveling salesman that needs to visit all the towns on a map within a minimal amount of time? Or a funny one about a wolf, a sheep and a cabbage that have to safely cross the river, one at a time, in a boat? One needs to strategize the boat trips given that it is not safe for the sheep to stay alone with the wolf on the shore or for the cabbage to be left with the sheep.

Image by Celine, distributed under the Creative Commons license

When we were back in school, we never imagined how applicable such training would become once we had kids. Just replace a wolf with your teenage daughter, a sheep with your toddler son, and a cabbage with your new baby, and everything starts to make sense. Remember that you can leave the baby at home with the teen but not the toddler, that your toddler has a playdate and both need boosters, that one should nap while two others are playing and the teen has a dance class, then one toddler has a swim lesson while you need to drop off his playmate and pick up your oldest one. And don't forget this doctor's appointment with the youngest one. Too easy? Then reorganize everything on-the-fly following a call from the nurse's office with a red terror signal coded “LICE” or “STREP.” Surprisingly we have all adapted to such daily optimization riddles, and masterfully manage them without any computerized support.

Here is a real life puzzles that may look painfully familiar:
Summer is not easy on the working parents. Camps are usually further away from home than schools are, many camps start later and end earlier than schools. So, go figure how to squeeze an 8-hour working day in-between...
This puzzle is inspired by my friend's real life story:


She lives 15 miles South from where she works (home marked H and work marked W on the sketch).
Her son goes to a tennis camp 10 miles West from home (marked C). She drops him off, then heads to work. As she is not able to pick her son from the camp at 4pm, she arranged a carpool with another family, that unfortunately lives in the opposite direction: 5 miles East from her (marked B on the sketch). So, on a regular camp morning, my friend leaves her house early to pick up this other kid, then drives both kids to the camp and then continues to work. In addition to all this, on one morning last week she got a call from a friend from work. Friend had some car trouble and pleaded to be picked up on her way to work. Friend lives 2 miles South from my friend's home (marked F).
What is the minimum number of miles my friend had to travel on this hectic morning before reaching her work?

See the answer.

Before letting anyone drive your kids, make sure they practice with a Wolf, a Sheep and a Cabbage:

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