Why it happens: the rule fixes every box above the bottom row, so once the bottom row is given the whole pyramid is decided — there is nothing left to choose. That is why a pyramid can be described completely by its bottom row alone.
Q2.
How do we fill this pyramid? (top 10; middle row 4, ?; bottom row 1, ?, ?)
Answer
Here the bottom row is not given, so read the rule backwards. “A box is the sum of the two below it” also says “a missing box below is the box above minus its known neighbour”.
Right box of the middle row: 10 – 4 = 6
Middle box of the bottom row: 4 – 1 = 3
Right box of the bottom row: 6 – 3 = 3
So the pyramid is: bottom 1, 3, 3; middle 4, 6; top 10.
The completed pyramid, filled by subtracting instead of adding.
Why it happens: if p + q = s and you know s and p, then subtracting p from both sides gives q = s – p. Subtraction is not a new rule — it is the same rule rearranged, and rearranging is legitimate because taking the same amount from both sides of an equation keeps it true.
What about filling in the numbers in this pyramid? Where do we start? (top 60; middle row ?, ?; bottom row 12, ?, 8)
Answer
You cannot start by subtracting — no box here has both the numbers it needs. So name the empty boxes with letter-numbers and let the rule give you equations.
The same pyramid with the three empty boxes named a, b and c.
a + b = 60
12 + c = a
c + 8 = b
Replace a and b in the first equation by what they equal:
(12 + c) + (c + 8) = 60
20 + 2c = 60
2c = 40 (subtracting 20 from both sides)
c = 20(dividing both sides by 2)
a = 12 + 20 = 32, b = 20 + 8 = 28
The completed pyramid: bottom 12, 20, 8; middle 32, 28; top 60.
Why it happens: substituting 12 + c for a is allowed because a and 12 + c are the same number — the second equation says so. Subtracting 20 from both sides and then dividing both sides by 2 keeps the two sides equal at every stage, because whatever you do to one side you do to the other. That is the only reason we are allowed to change an equation at all.