On day 30, it covers the entire pool for the first time.
On which day did it cover exactly half of the pool?
💡 Hint
Skip the area arithmetic and think in reverse: the rule that grows the patch also tells you how big it was when you step back in time.
Reveal answer
On day 29.
Step-by-step solution
- The patch doubles every day, so each day’s size is twice the size of the day before.
- Turn that around: each day’s size is half of the next day’s size.
- On day 30 the patch covers the whole pool. So on day 29 it covered half the pool.
- To see how lopsided doubling is, keep going back: day 28 is a quarter, day 27 an eighth, and day 25 is only 1/32, about 3% of the pool.
- Day 15, the tempting answer, would be 1/32,768 of the pool: a speck you could cover with a coin.
Why it fools people
Our intuition is linear. We picture growth as adding the same amount each day, so half the size should mean half the time. Doubling breaks that intuition, because the amount added each day keeps growing. In this pool, the algae covers less than 4% of the surface five days before the end, then fills the rest in the final stretch. The real answer feels too close to the deadline because our gut doesn’t expect growth to be so back-loaded.
Did you get it?
About This Puzzle
It’s mid-July, the family has been away at the lake, and nobody has checked the backyard pool in weeks. A small green patch appeared in one corner, and it has been growing in a very particular way. This puzzle is about doubling, the kind of growth that looks harmless for a long time and then takes over almost overnight. It’s an easy puzzle once you see it, but the first answer most people reach for is wrong by a wide margin. Read the numbers carefully, and pay close attention to what doubling means from one day to the next.
How to Think About It
Don’t start from the beginning of the month. Start from the end and work backward, because the end is the only day you know exactly. Ask a simple question: if the algae doubles every day, how does today’s patch compare with yesterday’s? Once you can say that in one sentence, apply it to the last day of the story. Working backward from a known endpoint is one of the most useful habits in puzzle solving, and it turns many hard-looking problems into one-step problems.
Common Traps
The most common mistake is to halve the number of days. That would be right if the algae grew by the same amount every day, like a fence being built one board at a time. But doubling is not steady growth: each day adds as much as all the previous days combined. Another trap is to try to work out the size of the original patch. You don’t need it, and the puzzle never gives it. Everything you need is in the relationship between neighboring days.
Where This Puzzle Comes From
Doubling puzzles are an old way to explain exponential growth. A well-known relative is the legend of the chessboard and the grains of rice, in which one grain on the first square doubles on every square after it. The pond version has long been used in classrooms and in discussions of population growth. The same shape appears in compound interest and in anything that spreads from person to person: small for a long time, then suddenly everywhere.
Try This Variation
Same pool, same algae, but this summer starts with two identical patches instead of one, each doubling every day. Before, a single patch filled the pool on day 30. On which day do the two patches together fill it?
Show the answer to the variation
On day 29. Two patches are exactly what one patch becomes after a single day of doubling, so the two-patch pool is always one day ahead. Starting with twice as much saves just one day, not half the time. With doubling, a doubled head start buys you exactly one step.