Concepts: Algebra Play (part 1)

1.1 Algebra as a tool for tricks

For two years algebra has helped us model situations and solve equations. This chapter uses it differently: to explain tricks and puzzles that look like magic. A trick that always gives the same answer is really an algebraic identity in disguise. Once the steps are written with a letter-number, the hidden constant appears.

Tricks also help us invent new puzzles. By changing one step and watching what happens to the final expression, we can design a trick with any answer we like. Number pyramids, calendar grids and divisibility tests follow the same pattern: first notice the result, then prove it with algebra.

Why letters instead of numbers? A letter stands for every possible starting value at once, so a single line of algebra replaces a whole table of examples. Each step is the same operation applied to the same unknown, and the simplification shows the unknown cancelling at the end. Inside the symbol, the number keeps changing while the letters stay fixed, which is exactly what a proof needs.

1.2 The think-of-a-number trick

The chapter's first trick has five steps. Think of a number, double it, add four, divide by two, and finally subtract the original number. The prediction is that the answer is always 2.

Write the starting number as x and follow the steps.

Start with the letter x. Doubling gives 2x. From there add four, obtaining 2x + 4. Halving that result produces x + 2. Finally take away the original letter, and only 2 remains.

The letter x cancels at the last step, so the answer cannot depend on the starting number. That is the entire trick: the steps are arranged to leave a constant. To make the final answer 3, add six instead of four in step 3; then the expression becomes x + 3 and the final answer is 3. To get 5, add ten.

1.3 The date trick

A friend thinks of a date written as day D and month M. The steps are: multiply the month by 5, add 6, multiply by 4, add 9, multiply by 5, then add the day. Following the algebra:

1. Month M.
2. Multiply by 5: 5M.
3. Add 6: 5M + 6.
4. Multiply by 4: 20M + 24.
5. Add 9: 20M + 33.
6. Multiply by 5: 100M + 165.
7. Add the day D: 100M + 165 + D.

The answer is 165 plus a two-digit day stuck after the month. To recover the date, subtract 165. The month is what remains before the last two digits and the day is the last two digits. For example, an answer of 291 gives 291 − 165 = 126, so M = 1 and D = 26: Republic Day. An answer of 1390 gives 1225, so the date is 25 December.

More dates: 1269 − 165 = 1104 gives 4 November; 394 − 165 = 229 gives 29 February; 296 − 165 = 131 gives 31 January. The trick works because 100M + D stores the month and day in separate digit positions, which is why the two-digit day is safe.

1.4 Inventing your own trick

A trick is designed backwards from the answer you want. Suppose the final answer should be 7. Start with x, double it to 2x, add 14, halve the result to get x + 7, and then subtract x. The final expression is 7 whatever number the volunteer chose. For any final answer k, add 2k before halving and subtract the original number after halving.

Change one step and the answer changes in a predictable way; that is how a trick maker moves from 2 to 3 to 5. Write the steps as expressions first, simplify, and read the constant off the result. The same design method works when the trick asks for a date or a digit rather than a single number. Keep each step reversible while you design it: a step that loses the starting information can make the trick impossible to explain even when the answer looks right.

1.5 Number pyramids

In a number pyramid, any entry is built by adding the two entries immediately beneath it. A two-row pyramid with bottom row a, b has top a + b. A three-row pyramid with bottom row a, b, c has middle row a + b and b + c, so the top is (a + b) + (b + c) = a + 2b + c.

That formula lets us find the top without building the pyramid. For bottom row 4, 1, 10 the top is 4 + 2 × 1 + 10 = 16. For 2, 5, 3 the top is 2 + 10 + 3 = 15. The middle number is counted twice because it feeds both boxes of the middle row. Every filled pyramid can be checked by adding upward step by step.