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A Carpenter's Puzzle

I came across this puzzle while reading Gelfand and Shen's splendid Algebra with my son. The book is full of many other gems, and it is a worthwhile purchase for any math geek.

Measure Twice, Cut Once?

You draw six lines on a stick of lumber to divide it fully into 7 equal lengths.

Then you change your mind and mark twelve lines to divide it up into 13 equal lengths.

Finally you change you mind one last time and cut the wood into 20 equal pieces.

When you are done, the two end pieces will have no marks on them. (Why?) But then you notice something interesting: the other 18 pieces all have exactly one mark on them. Some have a 1/7 mark, and some have a 1/13 mark, but never both.

Why does each inside piece end up with exactly one mark?

What would have happened if you had marked 35 pieces, then 53, then cut 35 + 53 = 88 pieces out of your lumber?

Continue reading if you would ike a hint....

A Soccer Player's Hint

I don't want to spoil things completely. Here is the answer in the form of a question.

We sometimes forget to bring a bottle of water to my son's soccer games.

So if he is lucky, the other kids will share their water with him.

If you have 6 bottles of water among 7 kids on your team and everybody shares equally, the fraction 6/7 tells you how much water you will get.

Suppose another team has a similar issue with 11 bottles and 13 kids. Everybody will get 11/13 of a bottle, just a little bit less water than the first team.

Now the two teams decide to join forces and share all their water equally. How much water does each person get once all the water is shared? Is that more or less than before?

Posted by David at August 6, 2008 11:03 PM