Earlier today, I set a puzzle by UCLA professor Terry Tao, who is widely regarded as one of the world’s top mathematicians. In an online game, you earn coins by harvesting from plants. There are two types of plant: a regular plant produces a single coin per day a mother plant produces a single coin and two seeds per day. (A seed will grow into a regular plant the following day.) At the start of the game, a player has a certain number of mother plants, regular plants and coins, but not more than nine of each.
Suppose that I begin on Day 1 with a mother plants, b regular plants and c coins. 1. How many coins do I have in total at the end of days 2 and 3? 2. Find an expression for how many coins I have at the end of Day X. 3.
At the end of Day 10 I have 427 coins. Day 3: 9a + 3b + c Workings: At the end of day 1 the number of coins is is a + b + c. Each mother plant has also produced 2 seeds, but these have not yet grown into regular plants.
On day 2, these plants produce a coin each, so there are 2a of these, added to another a from the mother plant and the a from day 1, makng 4a. The regular plants produce another coin each, making b in total to add to the b from day 1. This gets us 4a + 2b + c for day 2, and the same reasoning gets us the above expression for day 3.
We calculated for X = 1, 2 and 3. You hopefully saw the pattern emerge. The number of coins produced by the mother plant and her descendants is the square of the day. 100a + 10b + c = 427.
So a, b and c must be 4, 2 and 7. The answer was in the question! Tao said he suggested this puzzle because he likes the way it naturally introduces algebraic ways of thinking as the most efficient way to solve a problem.
Extract — continue reading at the source.
