2

Part of 1971 Bundeswettbewerb Mathematik

Problems(2)

Cutting paper into 8 or 12 pieces

Source:

You are given a piece of paper. You can cut the paper into

8

12

pieces. Then you can do so for any of the new pieces or let them uncut and so on. Can you get exactly

60

pieces¿ Show that you can get every number of pieces greater than

60

modular arithmetic

Error correcting codes: differing at three positions

Source: Bundeswettbewerb Mathematik 1971, round 2 problem 2

The inhabitants of a planet speak a language only using the letters

A

O

. To avoid mistakes, any two words of equal length differ at least on three positions. Show that there are not more than

\frac{2^n}{n+1}

n

combinatorics proposedcombinatorics