Puzzles

Published on March 15th, 2013 | by Simi

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Puzzle Challenge 2

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#1 INTEGER CHALLENGE

a.  Any $m+ 1$ distinct positive integers are chosen from the integers  $1,2,3 \dots, 2m$.  Prove that we can choose three integers $a, b$ and $c$ from the $m+1$ integers  (not necessarily distinct) such that $$a+b=c.$$

Construct an example showing that for $m$ integers this statement is not necessarily true.

b. Prove the similar result for $m+2$ distinct positive integers.

#2 EQUILATERAL TRIANGLE

An equilateral triangle is divided into $n^2$ ${small triangles}$. This is done by dividing each side into $n$ segments of equal length, and drawing straight lines through the resulting points parallel to the sides.

An example with $n = 21$ is shown below. There are 441 ${small triangles}$.

A sequence of little triangles is called a ${family}$ if no triangle appears more than once in it, and every triangle beginning with the second one has a common side with the previous triangle of the sequence. What is the maximum number of small triangles a family can have?

#3 FARMER FRED’S FENCES

Fred the farmer has forty three pieces of fencing whose lengths are $1, 2, 3 \dots,43$ metres.

a. Is it possible to fence a square field using all of these pieces?
b. Is it possible to fence a rectangular field using all of these pieces?

#4 CONGRUENT RECTANGLES

A rectangle $R$ has integer side lengths $m$ and $n$. Let $D(m,n)$ be the number of ways that $R$ can be cut into congruent rectangles with integer side lengths by means of cuts parallel to the sides of $R$ (with endpoints on opposite sides of $R$). Determine the perimeter $P$ of the rectangle for which $\frac{D(m, n)}{P}$ is maximised.

Puzzles contributed by Dr Michael Evans