AMC 8 · 2004 · #15

Easy mode Grade 4

Problem

Picture a shape made of hexagonal tiles. (Look at the figure below.) There are $13$ black tiles and $6$ white tiles.

Now we add a new border around the outside. The border is made of new white tiles, the same size and shape as the others. The border goes all the way around, with no gaps.

After the border is added, count the white tiles and count the black tiles in the new shape. What is the difference between the two counts?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: A figure is built from $13$ black and $6$ white hexagonal tiles. A border of new white hexagons (same size and shape) is added around the outside. After the border is added, what is (total white tiles) $-$ (total black tiles)?

Givens: Original figure: $13$ black tiles, $6$ white tiles; The figure is arranged so the white tiles form a ring around a central black tile, with another ring of black tiles around the whites; A new border of white hexagons is wrapped around the current outer edge; All hexagons are the same size and shape; Answer choices: (A) $5$, (B) $7$, (C) $11$, (D) $12$, (E) $18$

Unknowns: The difference (total white tiles) $-$ (total black tiles) after the new border is added

Understand

Plan

Primary tool: #5 Look for a Pattern

Secondary: #2 Make a Systematic List

The figure is built in concentric hexagonal rings: a center tile, then a ring of $6$, then a ring of $12$. The new border is just the next ring out. Tool #5 (Look for a Pattern) is the right primary — once we see the ring counts $6, 12, \ldots$ jumping by $6$, the next ring is forced to be $18$. Tool #2 (Make a Systematic List) lays the ring counts and colors in a small table so the final tally — new whites added, blacks unchanged — is one clean subtraction.

Execute — Answer: C

#2 Make a Systematic List 4.OA.C.5 Step 1

Identify the rings in the original figure.
A hexagon-tiled arrangement breaks into a center tile plus concentric rings.
Count what each ring holds and check against the given totals.

$\begin{array}{l|c|c} \text{ring} & \text{tiles} & \text{color}\\\hline \text{center} & 1 & \text{black}\\ \text{ring 1} & 6 & \text{white}\\ \text{ring 2} & 12 & \text{black} \end{array}$ \;black $= 1+12 = 13$, white $= 6$ \;(matches the problem)

💡 Grade 4 "generate a pattern" begins with reading off the structure. Putting the rings in a small table makes the next step automatic.

#5 Look for a Pattern 4.OA.C.5 Step 2

Read the ring-size pattern.
Ring $1$ has $6$ tiles, ring $2$ has $12$.
The jumps go $+6, +6, \ldots$ — each new ring around a hexagonal center adds another $6$ tiles.
So ring $3$ has $12 + 6 = 18$ tiles.

$$6, \;12, \;18, \;\ldots \;\;\text{(ring } n \text{ has } 6n \text{ tiles)} \;\Rightarrow\; \text{ring 3} = 18$$

💡 Grade 4 "identify apparent features of a pattern": the rule is "add $6$ each ring," which gives $18$ for the next ring without redrawing.

#2 Make a Systematic List 3.OA.D.8 Step 3

Update the color totals.
The new border is the third ring, and the problem says every new tile is white.
Black tiles are untouched.

$\text{black} = 13 \;\;\text{(unchanged)}$, $\;\text{white} = 6 + 18 = 24$

💡 Grade 3 multi-step word problem: track each color separately, then combine only what the question asks for.

#2 Make a Systematic List 3.OA.D.8 Step 4

Subtract to answer the question.
The problem asks for (total white) $-$ (total black).

$$24 - 13 = 11 \;\Rightarrow\; \textbf{(C)}$$

💡 One Grade 3 subtraction closes the problem — the pattern did the heavy lifting.

[1] #2 4.OA.C.5 Identify the rings in the original figure. A hexagon-tiled arrangement breaks in

[2] #5 4.OA.C.5 Read the ring-size pattern. Ring $1$ has $6$ tiles, ring $2$ has $12$. The jumps

[3] #2 3.OA.D.8 Update the color totals. The new border is the third ring, and the problem says

[4] #2 3.OA.D.8 Subtract to answer the question. The problem asks for (total white) $-$ (total b

Review

Reasonableness: Sanity check the count by a different route. Total tiles after the border: $1 + 6 + 12 + 18 = 37$. Of those, $13$ are black, so $37 - 13 = 24$ are white, and $24 - 13 = 11$ — matches (C). Quick elimination of distractors: (A) $5$ and (B) $7$ are too small (we added $18$ whites and no blacks, so the gap must grow by $18$ from the original $6 - 13 = -7$, landing at $+11$). (E) $18$ is the ring size itself, not the difference. (D) $12$ would need only $6$ extra whites, but the new ring has $18$. So (C) is the only consistent value.

Alternative: Tool #11 (Find an Invariant): black tile count is invariant under adding a white border, so the difference (white $-$ black) changes by exactly the number of new whites. The original difference is $6 - 13 = -7$; adding $18$ whites gives $-7 + 18 = 11$. Same answer (C), one line of arithmetic.

CCSS standards used (min grade 4)

3.OA.D.8 Solve two-step word problems using the four operations (Adding the new white tiles to the original whites and subtracting blacks to answer (white) $-$ (black).)
4.OA.C.5 Generate a number or shape pattern that follows a given rule; identify apparent features of the pattern (Reading the ring counts $6, 12, 18, \ldots$ and using the +$6$-per-ring rule to predict that the new border holds $18$ tiles.)

⭐ Each new hexagon ring grows by a fixed $6$ tiles — once you spot the rule, the new border's size is forced, and the rest is one subtraction.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

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