The clockwise ant

This puzzle comes from Argentinian puzzlist Jaime Poniachik, published in the February 1992 issue of Gamesmagazine. It is short enough to state in three sentences, and just slippery enough that most people's first instinct leads them somewhere wrong.

An ant crawls onto a clock face at the 6 mark, just as the minute hand is passing 12. She begins crawling counterclockwise around the face at a uniform speed. When the minute hand catches up with her, she reverses and crawls clockwise — same speed, opposite direction. Forty-five minutes after that first encounter, the minute hand catches her a second time, and she leaves. How much total time did she spend on the clock face?

The ant is faster than the minute hand. She laps it going one way, then gets caught coming back. The question is: how long did she travel before the first meeting?

The key insight that the puzzle deliberately obscures: the ant is faster than the minute hand. She doesn't wait to be caught — she runs counterclockwise and the minute hand eventually catches up to her from behind. Then she turns and runs clockwise, and the hand catches her again 45 minutes later.

Çözümü açmadan önce kısa bir durak: aşağıdaki sorular, bulmacanın neyi sorduğunu ve hangi bilinmeyenlerin önemli olduğunu netleştirir.

1. 1Between the two encounters, how many minute marks does the minute hand travel? What does that tell you about the time elapsed?
2. 2In that same interval, how many minute marks does the ant cover if she is moving clockwise and the hand catches her again? (Think in full laps plus remainder.)
3. 3From the start until the first meeting, the ant starts at the 6 and the hand at 12. How many minute marks separate them along the direction the ant crawls first?
4. 4Once you have a speed ratio between ant and hand, what single equation fixes the unknown time to the first encounter?

Take a moment with it before reading on. The puzzle has one clean trick at its center, and it clicks satisfyingly when found. The key is to think about the two phases of the ant's journey in terms of minute marks covered — not angles or distances.

The elegant move in this puzzle is measuring everything in minute marks rather than time. The clock face has 60 marks. The minute hand travels exactly one mark per minute. The ant's speed in marks per minute is unknown — that's what we need — but the ratio of speeds is accessible from the second phase of the journey, where we know both distances.

Between the two encounters, the hand moved 45 marks and the ant moved 105 marks. This 3:7 ratio is then imported into the first phase, where the ant and hand converge from opposite sides of a 30-mark gap (from 12 to 6 going the short way clockwise). The ratio constrains how far each must have traveled to meet, and the equation resolves to x = 9.

The puzzle is a system of two unknowns with two constraints. The second phase gives you the ratio. The first phase gives you the sum. From those two facts, everything follows.

Poniachik published this in 1992, and it has circulated steadily in puzzle collections since — partly because the answer (54 minutes) is specific enough to feel like a real answer, and partly because the counterintuitive speed relationship (the ant is faster, not slower) gives people something to argue about. The puzzle earns its place in the canon.

If you're in the mood for the opposite problem — a grid that punishes patience in weeks instead of minutes — there's Arto Inkala's hardest Sudoku, where the difficulty is all in the empty cells.