AMC 8 · 2000 · #9

Grade 6 arithmeticnumber-theory

Archetype Small-Domain Constrained Search

exponentsdigit-constraintssystematic-enumeration systematic-enumerationdigit-constraintscasework ↑ Prerequisites: exponentsmulti-digit-arithmetic

📏 Short solution 💡 3 insights 📊 Diagram

Problem

Three-digit powers of $2$ and $5$ are used in this "cross-number" puzzle. What is the only possible digit for the outlined square?
$\begin{array}{lcl} \textbf{ACROSS} & & \textbf{DOWN} \\ \textbf{2}.~ 2^m & & \textbf{1}.~ 5^n \end{array}$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: A small cross-number grid has two entries: $\textbf{1 DOWN}$ is a three-digit power of $5$, and $\textbf{2 ACROSS}$ is a three-digit power of $2$. In the grid, $\textbf{2 ACROSS}$ runs across the middle row of $\textbf{1 DOWN}$, so the middle digit of $\textbf{1 DOWN}$ equals the first digit of $\textbf{2 ACROSS}$. Find the digit in the outlined square — the units (rightmost) digit of $\textbf{2 ACROSS}$.

Givens: $\textbf{1 DOWN}$ is a three-digit power of $5$; $\textbf{2 ACROSS}$ is a three-digit power of $2$; The grid forces: middle digit of $\textbf{1 DOWN}$ $=$ first digit of $\textbf{2 ACROSS}$; The outlined square is the units digit of $\textbf{2 ACROSS}$; Answer choices: (A) $0$, (B) $2$, (C) $4$, (D) $6$, (E) $8$

Unknowns: The digit in the outlined square (units digit of $\textbf{2 ACROSS}$)

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #6 Guess and Check

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.

Execute — Answer: D

#2 Make a Systematic List 6.EE.A.1 Step 1

List the three-digit powers of $5$.
Multiply by $5$ until the result has four digits.
Only the values between $100$ and $999$ qualify.

$5^1=5,\ 5^2=25,\ 5^3=125,\ 5^4=625,\ 5^5=3125$. Three-digit: $\{125,\ 625\}$.

💡 The Grade 6 "whole-number exponents" skill: $5^k$ means multiply $5$ by itself $k$ times. Stop the list once you cross $1000$.

#2 Make a Systematic List 6.EE.A.1 Step 2

List the three-digit powers of $2$ the same way.

$2^6=64,\ 2^7=128,\ 2^8=256,\ 2^9=512,\ 2^{10}=1024$. Three-digit: $\{128,\ 256,\ 512\}$.

💡 Same idea, base $2$. There are exactly three candidates.

#6 Guess and Check 4.NBT.A.2 Step 3

Read the grid constraint.
The asy figure shows $\textbf{2 ACROSS}$ runs across the middle row of $\textbf{1 DOWN}$, so the first (hundreds) digit of $\textbf{2 ACROSS}$ equals the middle (tens) digit of $\textbf{1 DOWN}$.
Check the middle digit of each candidate for $\textbf{1 DOWN}$.

$125 \to \text{middle digit } 2$. $625 \to \text{middle digit } 2$. Either way: first digit of $\textbf{2 ACROSS} = 2$.

💡 Reading the tens place uses Grade 4 place value — and both possible $\textbf{1 DOWN}$ values agree, so the constraint is forced.

#6 Guess and Check 4.NBT.A.2 Step 4

Find which three-digit power of $2$ starts with $2$.
From $\{128, 256, 512\}$, only $256$ begins with the digit $2$.
So $\textbf{2 ACROSS} = 256$, and its units digit is the outlined square.

$$\textbf{2 ACROSS} = 256 \;\Rightarrow\; \text{units digit} = 6 \;\Rightarrow\; \textbf{(D)}$$

💡 Match the first-digit clue against the small list — only one power of $2$ survives.

[1] #2 6.EE.A.1 List the three-digit powers of $5$. Multiply by $5$ until the result has four di

[2] #2 6.EE.A.1 List the three-digit powers of $2$ the same way.

[3] #6 4.NBT.A.2 Read the grid constraint. The asy figure shows $\textbf{2 ACROSS}$ runs across t

[4] #6 4.NBT.A.2 Find which three-digit power of $2$ starts with $2$. From $\{128, 256, 512\}$, o

Review

Reasonableness: Confirm directly. $5^3 = 125$ or $5^4 = 625$ — both end with $\dots 2 \dots$ 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.

Alternative: Tool #11 (Find an Invariant): notice that every three-digit power of $5$ ends in $\dots 25$, so the middle digit is always $2$. That single observation pins the hundreds digit of $\textbf{2 ACROSS}$ to $2$ without listing both candidates — then a one-line check of $128, 256, 512$ still gives $256$ and the units digit $6$.

CCSS standards used (min grade 6)

6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents (Computing the three-digit powers of $5$ ($125, 625$) and of $2$ ($128, 256, 512$) by repeated multiplication.)
4.NBT.A.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form (Identifying the tens digit of a three-digit number ($\textbf{1 DOWN}$) and the hundreds and units digits of $\textbf{2 ACROSS}$ to apply the grid constraint and read off the outlined square.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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