Mental Floss

Time for Some Mental Hygiene: It's the mental_floss Weekly Puzzle

By Editorial Staff
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Cat videos are cool and all, but the next time you’re in need of a mid-day pick-me-up, consider something slightly more stimulating: our weekly brain teaser, created by the late, great mathematician and puzzle master Martin Gardner. This week, it’s up to you to figure out how to evenly divide a cake.

Once you've worked out a solution, scroll down to see if you're right.

From My Best Mathematical and Logic Puzzles, Martin Gardner, 1994, Dover Publications, Inc. Buy at Amazon.


There is a simple procedure by which two people can divide a cake so that each is satisfied he has at least half: One cuts and the other chooses. Devise a general procedure so that n persons can cut a cake into n portions in such a way that everyone is satisfied he has at least 1/n of the cake. 


Several procedures have been devised by which n persons can divide a cake in n pieces so that each is satisfied that he has at least 1/n of the cake. The following system has the merit of leaving no excess bits of cake. 

Suppose there are five persons: A, B, C, D, E. A cuts off what he regards as 1/5 of the cake and what he is content to keep as his share. B now has the privilege, if he thinks A's slice is more than 1/5, of reducing it to what he thinks is 1/5 by cutting off a portion. Of course if he thinks it is 1/5 or less, he does not touch it. C, D and E in turn now have the same privilege. The last person to touch the slice keeps it as his share. Anyone who thinks that this person got less than 1/5 is naturally pleased because it means, in his eyes, that more than 4/5 remains. The remainder of the cake, including any cut-off pieces, is now divided among the remaining four persons in the same manner, then among three. The final division is made by one person cutting and the other choosing. The procedure is clearly applicable to any number of persons.