A man fills a basket with eggs in such a way that the number of eggs added on…

Concept — growth by repeated doubling. When a quantity is increased each period by an amount equal to whatever is already present, it goes from x to x + x = 2x — it doubles every period. A quantity that doubles every period is a geometric progression with common ratio 2, so after n periods it equals (starting amount) × 2n. Read in reverse, every step backwards in time halves the quantity.

Application to this basket.

1. Let a be the number of eggs in the basket at the start. On any day the man adds exactly as many eggs as are already inside, so the content goes from x to 2x — the number of eggs doubles each day.

2. After n days the basket therefore holds a × 2n eggs: a geometric progression with first term a and common ratio 2.

3. The basket is exactly full on the 24th day, so its capacity is C = a × 224.

4. One-fourth full means the content equals C/4 = (a × 224)/4 = a × 224 − 2 = a × 222.

5. Equating the two expressions for the content on day n: a × 2n = a × 222, so 2n = 222 and n = 22 days.

Cross-check — count backwards from the full basket.

• On day 24 the basket is full. Undo one doubling: on day 23 it held half of the full amount.

• Undo one more doubling: on day 22 it held half of a half, that is one-fourth of the full amount — the same 22 days.

• The starting number a cancels out of the equation, so the result does not depend on how many eggs the man began with; only the number of doublings between the two states matters. One-fourth is 22 times smaller than full, i.e. exactly two doublings earlier, so 24 − 2 = 22.

Result. The basket was one-fourth full after 22 days.