How many primitive Pythagorean triples have hypotenuse less than 50? A triple is primitive when the three numbers share no common factor greater than 1.

Seven primitive triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29), (12,35,37) and (9,40,41).

Every primitive triple comes from a pair of coprime integers m > n of opposite parity, via a = m^2 - n^2, b = 2mn, c = m^2 + n^2.

Write \sigma(m) for the sum of the decimal digits of m. Among all integers n with 1 < n < 100, how many satisfy \sigma(n^2) = n, and what is the smallest of them?

Exactly one such n, namely n = 9, since 9^2 = 81 and 8 + 1 = 9.

Every number is congruent to its own digit sum modulo 9, so \sigma(n^2) = n forces n \equiv n^2 \pmod 9.

Factor the number 111111 completely into primes, and explain how the factorisation follows from the identity 111111 = \frac{10^6 - 1}{9} rather than from trial division.

111111 = 3 \cdot 7 \cdot 11 \cdot 13 \cdot 37.

10^6 - 1 factors as a difference of cubes *and* as a difference of squares — exploit both.

A drawer holds 10 red, 10 blue and 10 black socks, all loose and indistinguishable by touch. The room is completely dark. How many socks must you take to be certain of holding **two complete pairs**, where socks are paired up within a colour?

Failure is cheapest for an adversary who spreads the colours as unevenly as possible: three of one colour plus one of each of the others uses five socks and yields only one pair.

The n-th row of Pascal's triangle has entries \binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}. Find every n for which the sum of this row lies strictly between 1000 and 3000, and show why the sum has the form it does.

The row sum is 2^n, so the only rows in range are n = 10 (sum 1024) and n = 11 (sum 2048).

There is a one-line answer for the row sum, and you can get it by expanding (1+1)^n.

Take a standard 8 \times 8 chessboard and cut off two squares that sit at diagonally opposite corners. Can the remaining 62 squares be covered exactly by 31 dominoes, each covering two squares that share an edge?

Try it for a while and you will keep failing, but failure is not a proof. Look for a quantity that every domino affects in the same way.

At a meeting, 9 people shake hands with some of the others. Nobody shakes their own hand and the meeting is not necessarily complete. Show that the number of people who shook hands an odd number of times is even, and explain why a meeting of 9 people cannot have exactly 3 people shaking an odd number of hands.

The parity statement is forced: the number of odd-degree people is always even, so 3 is impossible.

Count the total number of handshake-ends rather than the handshakes.

What is the remainder when 7^{100} is divided by 5? Give the answer as an integer between 0 and 4.

Reducing the base modulo 5 is always allowed: 7 \equiv 2 \pmod 5.

An ant sits at one vertex of a solid wooden cube with side length 1. It wants to reach the vertex diagonally opposite, but it must crawl over the surface of the cube rather than through it. What is the length of the shortest such path?

A surface path looks curved in three dimensions, but any single face it crosses is flat. Try unfolding.

Three players each get a red or blue hat placed on their head. Each sees the other two hats but not their own. Simultaneously each player either guesses their own colour or passes; they agree on a strategy beforehand and win if at least one guesses correctly and nobody guesses wrongly. What is the best possible winning probability, and what strategy achieves it?

\frac{3}{4}, achieved by a parity strategy.

Count configurations rather than players: you need a rule that is correct on 6 of the 8 hat assignments and silent on the other 2.

You have two hourglasses: one runs for exactly 4 minutes, the other for exactly 7 minutes. You may flip either glass at any instant, including mid-run. You have no clock. How do you time an interval of exactly 9 minutes?

Start both glasses at once. Begin timing the moment the 7-minute glass first empties, and flip the 4-minute glass each time it empties. Stop when the 4-minute glass empties for the third time after that.

Start both glasses together. Once the 4-glass is flipped every time it empties, it fires on a strict 4-minute beat that is completely independent of the 7-glass.

Two railway stations are 100 kilometres apart. Train A leaves the first station at 40 km/h towards the second; simultaneously train B leaves the second station at 60 km/h towards the first. A fly starts on the front of train A and flies at a steady 90 km/h back and forth between the two trains, turning around instantly each time, until the trains meet. How far does the fly travel in total?

Summing the infinitely many legs of the fly's journey is possible but painful. Is there something you already know that makes it unnecessary?

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