AMC 8 · 2000 · #9

Grade 6 arithmeticnumber-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint
exponentsdigit-constraintssystematic-enumeration systematic-enumerationdigit-constraintscasework ↑ Prerequisites: exponentsmulti-digit-arithmetic
📏 Short solution 💡 3 insights 📊 Diagram
Problem
A small cross-number grid has two entries: 1 DOWN is a three-digit power of 5, and 2 ACROSS is a three-digit power of 2. In the grid, 2 ACROSS runs across the middle row of 1 DOWN, so the middle digit of 1 DOWN equals the first digit of 2 ACROSS. Find the digit in the outlined square — the units (rightmost) digit of 2 ACROSS.

Pick an answer.

(A)
0
(B)
2
(C)
4
(D)
6
(E)
8

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a Systematic List

There are only a few three-digit powers of each base, so Tool #2 (Make a Systematic List) gives us the complete set of candidates in seconds — multiply by 5 (or 2) until you leave the 100–999 range. Once both lists exist, Tool #6 (Guess and Check) handles the crossing constraint: the middle digit of the chosen 5-power has to match the first digit of the chosen 2-power. Listing both sides is faster and safer than any algebraic detour.

1STEP 1

Multiply 5 by 5 until you pass 999: the three-digit powers of 5 are 125 and 625.

5¹=5, 5²=25, 5³=125, 5⁴=625, 5⁵=3125. Three-digit: {125, 625}.
2STEP 2

The same way, the three-digit powers of 2 are 128, 256, and 512.

2⁶=64, 2⁷=128, 2⁸=256, 2⁹=512, 2¹⁰=1024. Three-digit: {128, 256, 512}.
3STEP 3

The grid ties 2 ACROSS's first digit to 1 DOWN's middle digit — and both 125 and 625 have middle digit 2.

125 → middle digit 2. 625 → middle digit 2. Either way: first digit of 2 ACROSS = 2.
4STEP 4

Among 128, 256, 512, only 256 starts with 2, so 2 ACROSS = 256 and the outlined square is its units digit.

2 ACROSS = 256 → units digit = 6 → (D)
Answer
6
Confirm directly. 5³ = 125 or 5⁴ = 625 — both end with … 2 … in the tens place, so the shared cell must be 2. Among the three-digit powers of 2 (128, 256, 512), the only one whose hundreds digit is 2 is 256. The outlined square sits at the units position of 256, which is 6. The answer is consistent no matter which of the two valid 5-powers you picked, which is exactly why the puzzle has a unique solution.
💡Key takeaway

When a puzzle hides a number behind a rule like "three-digit power of 5," list every option — there are usually only a few. Then the grid's shared cell does the rest of the work for you.

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