Terence Tao’s “Magic Money Plant” Puzzle Shows How Simple Games Reveal Algebraic Patterns

In a recent Guardian column, mathematician Terence Tao presented a puzzle based on an online game where players harvest coins from two kinds of plants. A regular plant yields one coin per day, while a mother plant yields one coin and two seeds each day; each seed becomes a regular plant the following day. Players start with *a* mother plants, *b* regular plants, and *c* coins (none exceeding nine). The column asked readers to determine the total coins after days 2 and 3, to derive a general formula for day *X*, and to find the initial values *a*, *b*, *c* when the day-10 total equals 427 coins.

The Guardian

By tracking daily production, the column showed that the coin count after day 2 is 4*a* + 2*b* + *c*, and after day 3 it is 9*a* + 3*b* + *c*. Observing the pattern, the author derived the general expression *aX² + bX + c* for the total coins at the end of day *X*. The article notes that this polynomial behaves like a three-digit number when *X* = 10, because 10² = 100, 10 = 10, and the constant term *c* combine to form the digits *a*, *b*, *c*.

The Guardian

Setting *aX² + bX + c* equal to 427 for *X* = 10 yields the equation 100*a* + 10*b* + *c* = 427. Solving gives *a* = 4, *b* = 2, *c* = 7. Substituting these values confirms the earlier day-2 and day-3 totals: 4·4 + 2·2 + 7 = 27 coins after day 2, and 9·4 + 3·2 + 7 = 49 coins after day 3.

The Guardian

The column ties the puzzle to themes in Tao’s popular-science book *Six Maths Essentials*, which explores how algebraic concepts underpin everyday situations.

The Guardian

By embedding a quadratic polynomial in a simple game, the puzzle demonstrates how abstract algebra can be visualized through everyday analogues. Readers are encouraged to see algebraic reasoning as a tool for problem-solving, aligning with Tao’s broader goal of making higher-level mathematics accessible to a general audience.

The Guardian