A million grains of sand is a heap. If we remove one grain of sand from this heap, we will still have a heap.
We can now keep repeating (2) until we only have a single grain of sand remaining.
Is this a heap? Clearly not. But what went wrong with our thinking?
This is called the Sorites paradox (soros being Greek for "heap") and is a classic paradox that has no real answer.
Both (1) and (2) are true, and we can indeed keep removing one grain of sand until we have a single grain remaining. If we remove one more grain, we're left with nothing, is this still a heap?
When does the heap become a non-heap?
?
Puzzle 18
Start with a number larger than 0, square it, add 4, double, take away 3, times 4 and finally subtract the original number.
If you were now left with 20, what number did you start with?
Reasoning
If we convert the question to algebra, we have:
((n^2 + 4) x 2 − 3) x 4 − n = 20
Expanding the brackets and simplifying gives:
(2n^2 + 8 − 3) x 4 − n = 20
(2n^2 + 5) x 4 − n = 20
8n^2 + 20 − n = 20
8n^2 − n = 0 (*)
8n − 1 = 0
8n = 1
n = 1/8
In the equation marked (*) zero is also a potential solution, but as the question tells us that we "Start with a number larger than 0" we know that n can't be 0, and therefore we can safely divide by n.
Reasoning There are 61 squares of size 1 x 1. There are 37 squares of size 2 x 2. There are 15 squares of size 3 x 3. There are 3 squares of size 4 x 4.
