Problems

A \(5\times5\) grid is given with \(25\) coins, each red on one side and white on the other. coins are placed on the grid one at a time. When a counter is placed, any coins on neighbouring squares are flipped. Two squares are neighbours if they share a side (not just a corner). The aim is to finish with all coins showing red. During the process coins may flip several times. How many flips occur in total?

The entire plane is coloured using two colours: red and blue. Prove that there must exist two points of the same colour that are exactly \(1\) meter apart.

Give a visual proof that the sum of consecutive numbers until \(n\), i.e: \(1+2+\cdots + n\), where \(n\) is some whole number; is equal to \(n(n+1)/2\).

Use a visual proof to find the value of \[\frac{1+3+5+\cdots +2n-1}{(2n+1)+(2n+3)+\cdots + (4n-1)}\] You are not allowed to use the result from the examples to simplify the fraction.

The Pythagorean Theorem is a very useful tool in geometry. It says that if you have a right-angled triangle with sides measuring \(a,b,c\) where \(c\) is the longest side of the three, then \(a^2+b^2=c^2\). Explain how the following diagram gives a visual proof of the Pythagoraen Theorem.

Without carrying out the multiplications, which is larger: \[(2015+2026)^2\qquad \text{or} \qquad 4\times 2015\times 2016\]

Give a visual proof of the following identity \[(1^1\times 1!)\times (2^2\times 2!)\times (3^3\times 3!)\times\cdots \times (k^k\times k!)=(k!)^{k+1}\]

By cleverly dividing a square of side length \(1\), show that the sum \[\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots + \frac{1}{1024}= \frac{1023}{1024}\]

For natural numbers \(n\) and \(k\) with \(k\leq n\), the notation \({n\choose k}\) means the number of ways one can choose \(k\) objects from a set of \(n\) objects. Explain how the diagram below gives a visual proof of the fact that \[{n+1\choose 2}={n\choose 2}+n.\]

For a real number \(x\), we call \(|x|\) its absolute value. It is defined as whichever is larger: \(x\) or \(-x\). For example, \(|-2|=2\) and \(|3|=3\).

One of the most important inequalities involving absolute values is the triangle inequality, which states that \[|a+b| \le |a| + |b|.\]

Show that this inequality is true.