Waka-Waka. Your turn to play.

It is a special treat for me when you share your own life-math puzzles, and a pleasurable challenge if you ask me to help you solve them.
Life is a Puzzle. Let's solve it with numbers.
We have a wonderful community of readers, with diverse backgrounds and with various experiences, and together we can crack almost any problem.
Case in point: Last week, a girls' soccer coach and a reader, Lauren, asked me to help creating a play/rest timetable for her team. This is a fun real-life math challenge to share with your family and friends.

Image from Flckr distributed under the CCL

Lauren wrote:
We have 11 girls on our soccer team.
The games are split into 2 halves, each one is 30 minutes long for a total of 60 minutes in a game.
3 of our girls play goalie
There are 5 girls on the field and 1 goalie per shift for of a total of 6 girls on the field at one time.
The goalies should play equal amounts of the game and play consecutive shifts without subbing until their shift is over. After playing goalie or before playing goalie, the girl should be on the field as one of the 5 playing.
Each of the other 8 girls on the team who are not playing goalie, should play equal shifts on the field, alternating being in the game and being on the sidelines.
How many shifts do the 8 girls get and how long should the shifts be?
Also, how many shifts do the 3 girls get in goal and how long should their goalie shifts be?
Is there a solution to this problem?

As Shakira is singing to the World Cup players:
The pressure is on
You feel it
But you've got it all
Believe it!

Let's see. We have 60 minutes in a game overall.
At any given moment there are 6 players on a field. So, we have 60 x 6 = 360 playminutes total to divide among players.
60 of these minutes are goalie minutes and the rest 300 are regular playing minutes.

Start with dividing goalie time, as it is easier:
60 mins, 3 goalies, to be fair we need 60/3 = 20 mins for each goalie at the goal.

Now, we have 300 regular playing minutes. 8 players that should get equal amounts of time on the field and plus it seems that we should leave some play time for the goalies to run around with a ball. Lauren doesn't say whether each of 3 goalies should play the same time as regular players, so we assume that this is not a requirement.
Assume that regular players each play time T1 and goalies each spend T2 min running in the field.
So we have:
8 x T1 + 3 x T2 = 300

There are actually many solutions to this.
Pick any reasonable number of minutes as T1 and get a matching value for T2 from this equation.
The easiest will be to let each of the 8 regular players play a full half = 30min.
If T1=30min
8 x 30 + 3 x T2 = 300
240 + 3 x T2 = 300
3 x T2 = 60
T2 = 20

Each of the 3 goalies will then spend 20 min at the goal, 20 min playing in the field and 20 min resting on the sidelines.
Each of the 8 regular players will then spend 30 min playing. 4 will play first half and 4 second half.
Easy and fair.

But what will happen if somebody gets sick? How should coach Lauren re-adjust the calculations on a day of the game based on the show up? Assume we have G goalies and P regular players. How can we divide the game time using coach Lauren's description of fair play? Submit your thoughts in the blog post comment.

Image from Flckr distributed under the CCL

Aside from sport, similar scheduling tactics is applicable to many other domains such as dividing a time share property, traffic control, shift management at the restaurant and even chores at home.


  1. Hopefully we'll see game based learning improve! From the game to learn team

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