The cut that moves nothing
A hall has 8 rows of 25 chairs. A gap cuts every row into 20 chairs and 5 chairs. How many chairs are there?
Step 1 of 4
- A block, 8 rows of 25 chairs.
- A gap cuts it: 20 chairs, then 5.
- Left block: 8 × 20 = 160.
- Right block: 8 × 5 = 40. Total: 200.
Cutting a block changes nothing
There are 200 chairs. The whole block has 8 × 25 = 200. The two parts have 160 + 40 = 200. The count is the same wherever you cut. Every cut gives two parts. The two parts always add back to the whole.
Here are three different cuts of the same block.
| Gap after | Left | Right | Total |
|---|---|---|---|
| chair 20 | 8 × 20 = 160 | 8 × 5 = 40 | 200 |
| chair 13 | 8 × 13 = 104 | 8 × 12 = 96 | 200 |
| chair 1 | 8 × 1 = 8 | 8 × 24 = 192 | 200 |
Take a block of 24 rows by 31 chairs. Add one more column, a chair in every row. The count grows by 24, one chair for each row. So 24 × (31 + 1) = 24 × 31 + 24 = 744 + 24 = 768.
Slide the gap. Both parts still add to 200.
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