Brainteasers & Puzzles

Lily Pads on a Pond

Twenty-seven lily pads, each one square foot, sit on a six-thousand-square-foot pond, and every pad doubles in area each day. How many whole days until the pads blanket the pond?

solvedeasy1 min

A pond of area 60006000 square feet holds 2727 lily pads, each starting at 11 square foot. Every pad doubles in area each day. After how many whole days is the combined area of the pads at least the area of the pond?

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#Combined area after d days

Each pad starts at 11 square foot and doubles daily, so after dd days a single pad has area 2d2^d square feet. The number of pads never changes, so their combined area is

A(d)=27⋅2d.(1)A(d) = 27 \cdot 2^d. \tag{1}

#The covering condition

The pads blanket the pond once A(d)≥6000A(d) \ge 6000, which rearranges to

2d≥600027=20009≈222.2.(2)2^d \ge \frac{6000}{27} = \frac{2000}{9} \approx 222.2. \tag{2}

#Reading off the day

Powers of two bracket that threshold,

27=128<222.2≤256=28,(3)2^7 = 128 < 222.2 \le 256 = 2^8, \tag{3}

so the smallest integer dd that satisfies the condition is d=8d = 8. Checking the two boundary days confirms it.

Day ddCombined area 27⋅2d27 \cdot 2^d
7734563456
8869126912

On day 77 the pads cover 3456/6000≈0.583456 / 6000 \approx 0.58 of the pond, still a little over half, and one day later they overflow it. That is the doubling twist, the jump from barely-half to fully covered happens in the final step.

The pond is covered on day 88.