AMC 10 · 2022 · #6
Grade 5 arithmeticPick an answer.
Counting blocks one by one for 100 blocks is grinding. Tool #16 (Change Focus) flips the question: instead of counting blocks, count the multiples of 7 from 1 to 1000 and notice how they distribute across blocks. Tool #15 (Reorganize) groups multiples by their host block — each block has either 1 or 2 multiples (never 0 or 3+ in a window of 10). Tool #9 (Easier Problem) anchors why: spacing between multiples of 7 is 7, and a block of width 10 can fit at most two. So if we know the total count of multiples and the minimum per block, the leftover tells us how many blocks have an extra one. No algebra needed.
Look at the first few blocks
The first blocks hold one or two.
Looking at three small blocks shows that each block always has either one or two multiples — never zero.
4.OA.B.4Solve An Easier Related ProblemBound one block
No block can hold three.
Spacing of 7 inside a window of 10 forces exactly one or exactly two — counting by spacing instead of by block.
A spacing of seven inside a window of ten forces exactly one or exactly two multiples.
▸ Why?
Only the remainder after dividing decides where the multiples fall inside the window.
▸ Why?
A window wider than the spacing must catch at least one, and one too narrow for three catches at most two.
Count all the multiples
Count all the multiples of seven.
Count the multiples directly — that is the 'flipped' question.
5.NBT.B.6Change Focus Count The ComplementSet up two equations
Block count and multiple count give two equations.
Pretend every block had one multiple — that accounts for 100. The extra 42 multiples are the 'surplus', one per block that has two.
4.OA.A.3Organize Information In More WaysSolve and read the answer
Solving gives 42.
Reading the surplus directly as the answer — the change of focus pays off.
4.NBT.B.4Change Focus Count The ComplementThis AMC 12 problem only needs Grade 5 division and the multiples idea you already know — there are 142 multiples of 7 up to 1000 spread across 100 blocks, every block has at least one, so the extra 142 - 100 = 42 tells you exactly how many blocks have two.