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.

Recreational mathematics puzzles with complete proofs

Recreational mathematics, three tiers deep

Puzzles with proofs, not just answers

Issue 28 16 Oct 2026 28 puzzles published

This issue

New puzzles are added every Monday, Wednesday and Friday.

28 Primitive Triangles With a Short Hypotenuse
27 The Square Whose Digits Add Back to the Root
26 The Six-Digit Repunit
25 Socks in the Dark
24 Which Row of Pascal's Triangle Sums to a Given Range

Warm-up

16 published

One idea, a minute or two. Read the statement twice and the trick is usually visible.

01 Eyes and Legs in the Yard
02 When the Hands Sit on Top of Each Other
04 Dipping a Cube in Paint
05 How Many Zeros Hides 100!
07 Four Travellers and One Torch
09 The Frog That Counts Wrong

Warm-up — All issues →

Core

11 published

Two or three moves chained together. These are the ones worth writing down.

03 Five Points in a Unit Square
06 At Least One Is a Boy
08 Stamps of 3 and 5
10 A Knight That Comes Home
12 How Often Does a Perfect Shuffle Repeat?
14 Area Without Integration

Core — All issues →

Olympiad

1 published

Competition-shaped. The first move is rarely the obvious one; the proof carries the weight.

15 Writing 2026 as a Sum of Two Squares

Olympiad — All issues →

Slate & Sum

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Warm-up

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