Chapter 13: Algebra Play
Algebra is most famous for solving for the unknown. But it has another, more playful, side: it can explain why a magic trick always works. In this chapter we use algebra not as a tool to find a single answer, but as a key that unlocks patterns.
We will see how any "think of a number, do these steps, tell me the result" trick can be cracked by writing the starting number as and watching what happens. We will fill in number pyramids where each cell is the sum of the two beneath it. We will pluck four numbers from a calendar and use algebra to recover them from just their sum. And we will hunt for the largest possible product when we have a fixed set of digits to arrange.
The thread tying it all together is this: behind almost every "magic" trick lurks a simple algebraic identity. Once you see the identity, the magic becomes maths.
By the end you will see algebra as a bridge between concrete numbers and the general patterns they obey , and you will be able to invent tricks of your own.
What's inside
- Think-of-a-number tricks , how letter-numbers explain the magic.
- Number pyramids , sums climb the pyramid; algebra walks back down.
- Calendar magic , guess four dates from one sum.
- Algebra grids and shape puzzles , solving for hidden values.
- The largest product , arranging digits to maximise a product.
Key results
| Idea | Formula |
|---|---|
| Linear identity | (the "double, add , halve, subtract" trick) |
| Pyramid top (3 rows) | from bottom |
| Pyramid top (4 rows) | from |
| calendar sum | from corner |
| Largest product (digits ) for | put two largest digits first in tens places |
A calendar fact. In any month's table, moving down one row adds to the date, and moving right one column adds .
How to read this chapter
Read with a pencil. Every trick in this chapter has a hidden equation; try to predict the algebra before peeking at the explanation. Once you can do that, you can invent a new trick of your own.