Legend says there was once a temple in India where a brass plate held 64 golden discs stacked on 3 diamond needles. The priests were given one task: move all 64 discs from one needle to another.
A French mathematician, inspired by this legend, invented the Towers of Hanoi puzzle. It is also called the Tower of Brahma.
Let's try it with 3 discs first.
Tap a rod to pick up its top disc, then tap the rod you want to move it to. You can also drag.
Level 1 of 3
| Discs | 3 | 4 | 5 |
|---|---|---|---|
| Fewest moves | ? | ? | ? |
| Discs | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| Moves | 7 | 15 | 31 |
Write the fewest moves needed for 6 and 7 discs.
| Discs | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| Moves | 7 | 15 | 31 | 63 | 127 |
Write x² as x^2. For example, 3^n.
| Discs | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| 23−1 | 24−1 | 25−1 | 26−1 | 27−1 | |
| Moves | 7 | 15 | 31 | 63 | 127 |
If the number of discs is n, the fewest moves is 2n − 1.
It was said that when the priests finish moving all 64 discs, the world will end. How long would the priests take?
In another distant land, a king wanted to reward the inventor of chess. The inventor asked only for rice: 1 grain on the first square of the board, 2 on the second, 4 on the third, and so on, doubling on every square up to the 64th.
The king agreed. It sounded like a small ask. Let's see what he had agreed to.
| Square | Grains |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 4 |
| 4 | 8 |
| 5 | |
| 6 |
Write the grains on the 5th and 6th squares.
On the nth square, the number of grains is
Type the base in the big box and the power in the small raised box.
The nth square holds 2n−1 grains. After n squares, the total on the board is 2n − 1 grains.
That is the same number as the fewest moves for n discs. Both double at every step. This is exponential growth.
The inventor had asked for 264 − 1 grains of rice. Written out, that is
About 18 billion billion grains, roughly 1,000 times the rice the whole world grew in 2010.
And the priests? Moving one disc every second, they would need about 585 billion years. The universe is only about 14 billion years old.
That is the power of exponential growth.