AMC 10 · 2010 · #25
Grade 7 number-theoryPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Going forward is hopeless: we would have to test starting numbers one by one until one happens to take 7 subtractions. The end of every list is the same — it is always 0 — so Tool #11 (Work Backwards) fits perfectly. We start at 0 and grow the list one term at a time, and at each stage we ask for the smallest number that could sit above the current one. To answer that we need Tool #4 (Introduce a Variable): call the square that gets subtracted s² and turn the phrase 'largest square not exceeding the value' into an inequality on s. Tool #14 (Extreme Principle) is what lets us trust the greedy choice — because the smallest predecessor grows when its target grows, taking the smallest option at every step really does give the smallest N overall. Finally Tool #3 (Eliminate Possibilities) closes it: only one answer choice matches the units digit we compute.
Count steps, not numbers
8 numbers means 7 subtractions. Every list ends at 0, so start at 0 and build upward, keeping each new row as small as possible.
The finish line 0 never changes, so it is far easier to grow the chain up from 0 than to hunt for a starting number.
6.EE.B.5Work BackwardsRule for the smallest predecessor
If a row holds c, the row above is c+s². For s² to stay the largest square, c+s² must be under (s+1)², so s is at least half of c.
If the subtracted square were too small, the leftover would spill past the next square and a bigger square would get used instead.
7.EE.B.4Introduce A VariableBuild the chain upward
Apply the rule seven times, always adding the smallest legal square: 1,1,1,4,16,144,7056. The chain climbs 0,1,2,3,7,23,167,7223.
Each jump is forced to be the tiniest step that keeps the same subtracted square, so the ladder rises as slowly as it possibly can.
Each jump is forced to be the tiniest step that still keeps the same subtracted square.
▸ Why?
A smaller step would spill past the next square, so a different square would be used instead.
▸ Why?
Bigger targets always need bigger predecessors, so picking the smallest at each step can never be beaten.
Why 7223 is truly the smallest
A bigger target always needs a bigger predecessor, so taking the smallest at every row gives the smallest top number, 7223.
Because bigger targets need bigger predecessors, greedily picking the smallest each step can never be beaten.
6.EE.B.5Extreme PrincipleRead the units digit
Only the units digit is asked, and 7223 ends in 3 — exactly one answer choice matches.
Only choice (B) ends in 3, so every other option is ruled out at a glance.
6.EE.B.5Eliminate PossibilitiesBuild the chain up from 0 and always add the smallest square that keeps the same subtraction — the top of the shortest-growing ladder is the smallest starting number.
- Count steps, not numbers
- Rule for the smallest predecessor
- Build the chain upward
- Why 7223 is truly the smallest
- Read the units digit