13.8 Summary
Key Points
- Algebra models and explains numerical situations, so it surfaces across nearly all of mathematics, science and beyond.
- A “think of a number” trick gives everyone the same answer exactly when the chosen number cancels from the final expression. Writing the steps in x exposes (and lets you design) this cancellation.
- The date trick produces 100M + 165 + D; subtracting 165 recovers the month M and day D from the answer’s digits.
- In a number pyramid each box is the sum of the two below. Naming unknowns with letters turns “stuck” pyramids into solvable equations; the apex of a base a,b,c is a + 2b + c, and of a,b,c,d is a + 3b + 3c + d (Pascal’s-triangle weights).
- A 2\times 2 calendar block has sum 4a + 16, so its four numbers can be recovered from the sum alone. Shape grids are systems of equations written as pictures.
- To maximise (two-digit) \times (one-digit): use the largest digit as the multiplier, the other two in decreasing order.
- Place value unlocks divisibility tricks: swapping a 2-digit number changes it by 9(b-a); abc + bca + cab = 111(a+b+c) = 3\times 37\times(a+b+c); and abcabc = abc \times 7 \times 11 \times 13.
- Algebra is an indispensable tool for justifying mathematical statements, and a wonderful toy for inventing puzzles of your own.