Penny Doubled Daily Calculator

The classic “penny doubled every day for 30 days” puzzle is the best demonstration of exponential growth anyone has come up with. It is also the thing most people get wrong about investing, compound interest, and how wealth actually accumulates.

The Famous Example

Would you rather have $1,000,000 today, or a penny doubled every day for 30 days?

Most people instinctively choose the million dollars. The math reveals why that is the wrong choice:

A penny doubled daily for 30 days becomes over $5.3 million, more than five times the million dollar alternative. Look at where it happens, though: on day 25 you still have $167,772, and the last five days alone add $5.2 million. Almost all of the growth arrives at the end.

The Formula

Day 1 is the starting amount, before any doubling. So for a starting amount A on day n:

Final Value = A × 2^{(n − 1)}

That exponent is the part people get wrong. A penny for 30 days is 0.01 × 2²⁹ = $5,368,709.12, not 0.01 × 2³⁰. Using n instead of n − 1 doubles your answer, and it is the single most common error in write-ups of this puzzle.

More generally, for any rate rather than a straight doubling: Final Value = A × (1 + rate)^{(n − 1)}

Why This Matters for Investing

This example illustrates why compound interest is so powerful in long-term investing. While no investment doubles every day, the principle holds:

• 7% annual return doubles every 10 years or so (Rule of 72: 72 ÷ 7 = 10.3)
• 10% annual return doubles every 7 years
• 14% annual return doubles every 5 years

Starting early beats starting bigger, and it is not close. The person who invests a small amount at 25 and stops usually finishes ahead of the one who starts at 40 and invests more, because the first one bought extra doublings and doublings are what the whole thing runs on.

The Rule of 72

To quickly estimate how long it takes to double money at a given interest rate: Doubling Years ≈ 72 ÷ Annual Interest Rate (%)

Example: At 6% per year, money doubles in 72 ÷ 6 = 12 years.