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.

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.

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.

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.

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?

You have nine coins that look identical. Exactly one of them is heavier than the others; the other eight weigh the same. Using a balance scale with two pans, what is the fewest number of weighings that is guaranteed to identify the heavy coin, whatever the outcome?

Each weighing has three possible outcomes, so k weighings can distinguish at most 3^k situations.

A single lily pad doubles the area it covers every day. Left alone, the patch covers an entire pond in 48 days. On which day is the pond exactly half covered?

Day 47 — the pond is half covered with only one doubling left to go.

You do not need to know the size of the pond or of one pad.

What is the largest number of pieces you can divide a round pizza into using exactly six straight cuts with a long knife? You may not stack or move the pieces.

The n-th cut adds as many new pieces as the number of existing regions it passes through.

A frog at the bottom of a dry well 20 metres deep climbs 3 metres during each day, then slips back 2 metres each night while it sleeps. It starts climbing on the morning of day 1. On which day does it first reach the rim?

The net gain of 1 metre per day is a trap: the frog leaves the well during a climb, not at the end of a night.

Four hikers must cross a rickety bridge at night. They have one torch, the bridge holds at most two people at a time, and anyone crossing must carry the torch. A pair walks at the speed of its slower member. Individually they need 1, 2, 5 and 10 minutes. What is the shortest total time in which all four can be across?

The naive plan — fastest person shuttles everyone — costs 2 + 1 + 5 + 1 + 10 = 19 minutes. Can the two slowest travellers be made to cross together?

The number 100! = 1 \cdot 2 \cdot 3 \cdots 100 ends in a long run of zeros. How many trailing zeros does it have — that is, what is the largest k such that 10^k divides 100!?

Every factor of 10 needs one 2 and one 5. Which of the two runs out first in 1 \cdot 2 \cdots 100?

A wooden cube 4 units on a side is dipped in paint so that every outer face is covered, then sawn into 64 unit cubes. You reach in and draw one small cube at random. What is the chance that it shows painted paint on **exactly two** of its faces?

\dfrac{24}{64} = \dfrac{3}{8}, so a probability of 0.375.

Classify the little cubes by how many of their faces were on the outside: corners, edge cubes, face centres, and the hidden interior.

Sometime between 3{:}00 and 4{:}00 the hour hand and the minute hand of an ordinary clock lie exactly on top of one another. At what time does that happen, to the nearest second?

About 3{:}16{:}22 — precisely 16\frac{4}{11} minutes past three.

Give both hands a speed in degrees per minute and track the angle between them as a function of t.

A walled yard contains only sparrows and cats, and you may not look inside. Counting over the wall you get 40 eyes and 56 legs. Every animal is intact: no one-legged sparrows, no three-legged cats, and at least one of each species is present. How many cats are in the yard?

**8 cats**, and therefore 12 sparrows.

Two eyes per animal is not a coincidence — the eye count alone gives you the total number of animals before you touch the legs.

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Warm-up

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

16 published

27 The Square Whose Digits Add Back to the Root 14 Oct 2026
25 Socks in the Dark 9 Oct 2026
23 A Chessboard with Two Corners Missing 5 Oct 2026
21 The Last Digit of a Colossal Power 30 Sep 2026
20 An Ant Walks Around a Cube 28 Sep 2026
18 Nine Minutes from a 4 and a 7 23 Sep 2026
17 Two Trains and a Persistent Fly 21 Sep 2026
16 Nine Coins, One Counterfeit 18 Sep 2026
13 The Pond That Fills Overnight 9 Sep 2026
11 Six Straight Cuts of a Pizza 3 Sep 2026
09 The Frog That Counts Wrong 29 Aug 2026
07 Four Travellers and One Torch 23 Aug 2026
05 How Many Zeros Hides 100! 17 Aug 2026
04 Dipping a Cube in Paint 15 Aug 2026
02 When the Hands Sit on Top of Each Other 9 Aug 2026
01 Eyes and Legs in the Yard 6 Aug 2026

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