AMC 10 · 2016 · #25
Grade 9 number-theoryPick an answer.
The blackboard game is really one formula, b(x)=⌊ x²/10^k ⌋, so tool #4 (Introduce a Variable) is the engine twice over: first to name b(x), then to recenter with x = 5 · 10^k-1+m, which is the move that makes everything collapse. Tool #9 (Solve an Easier Related Problem) supplies the picture: play the k=2 game by hand, watch where it stumbles, and the mechanism for general k is visible. Tool #14 (Extreme Principle) does the two minimality arguments the problem hinges on — the earliest x at which a jump of 2 is even possible, and then the exact first x where it happens. Tool #5 (Look for a Pattern) finishes: the 1008 values of f stack into two repeating digit strings, and the digit sum falls out of a column count.
Turn erasing into floor division
Erasing is a plain division with rounding.
Chopping off the last k digits is the same as dividing by 10^k and keeping only the whole-number part.
5.NBT.A.2Introduce A VariableThe board is an unbroken run
The board records an unbroken run.
A counter that only ever ticks up by 0 or 1 leaves no holes, so the first hole sits one step past the last value before the jump.
A counter that only ever ticks up by zero or one leaves no holes, so the first gap sits just past the last jump-free value.
▸ Why?
Consecutive values differ by the same small fixed amount, so nothing between them can be skipped.
▸ Why?
Chopping off the last digits is dividing and keeping the whole part, and that quotient climbs only when a full block is filled.
Play the k=2 game by hand
The smallest case is played by hand.
Past x = 50 the board is really just counting x - 50, and it stumbles exactly when the leftover square (x-50)² first reaches 100.
8.EE.A.1Solve An Easier Related ProblemA jump needs 2x-1 ＞ 10²ⁿ
A jump needs the gap to exceed the divisor.
To leap over a whole block of size 10²ⁿ, consecutive squares have to be more than 10²ⁿ apart, and their gap is only 2x-1.
9.A-CED.A.1Extreme PrincipleRecenter at 5 · 10²ⁿ⁻¹
Recentring makes the formula clean.
Measuring x from 5 · 10²ⁿ⁻¹ turns the board into a counter m plus a small square that only occasionally spills over.
9.A-SSE.A.2Introduce A VariableThe first spill-over gives f(2n)
The first spill-over names the missing value.
The counter ticks up by one every turn until the leftover square finally fills a whole 10²ⁿ, and that single moment is the double tick.
8.EE.A.1Extreme PrincipleSplit the total into two stacks
The long total splits into two stacks.
Every f is the same two shapes shifted along, so stacking all 1008 of them makes two clean repeating digit strings.
8.EE.A.1Look For A PatternAdd the columns and total the digits
Adding the columns gives 8064, choice (E).
No column reaches 10, so nothing carries and the two digit sums simply add.
5.NBT.A.1Look For A PatternErasing k digits is dividing by 10^k, and past x = 5 · 10^k-1 the board just counts upward until the leftover square finally fills a whole 10^k — that one moment is the number it skips, and stacking all 1008 of those numbers adds column by column with nothing carrying.
- Turn erasing into floor division
- The board is an unbroken run
- Play the k=2 game by hand
- A jump needs 2x-1 ＞ 10²ⁿ
- Recenter at 5 · 10²ⁿ⁻¹
- The first spill-over gives f(2n)
- Split the total into two stacks
- Add the columns and total the digits