AMC 10 · 2013 · #13

Grade 4 counting

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Periodic Placement Formula

triangular-numberspattern-recognitionsequences-arithmetic pattern-recognitionidentify-subproblems ↑ Prerequisites: pattern-recognition

📏 Short solution 💡 2 insights

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Problem

Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying $``1"$ , so Blair follows by saying $``1, 2"$ . Jo then says $``1, 2, 3"$ , and so on. What is the $53^{\text{rd}}$ number said?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: Jo and Blair take turns counting up from 1. On each turn a person recounts from 1 up to one more than the last number the other person said. Jo says "1", Blair says "1, 2", Jo says "1, 2, 3", and so on. Find the 53rd number spoken overall.

Givens: Turn 1 (Jo): says 1 number, the list 1; Turn 2 (Blair): says 2 numbers, the list 1, 2; Turn 3 (Jo): says 3 numbers, the list 1, 2, 3; Every turn restarts the count from 1 and goes one higher than before

Unknowns: The value of the 53rd number said, counting every spoken number in order

Understand

Plan

Primary tool: #5 Look for a Pattern

Secondary: #7 Identify Subproblems, #2 Make a Systematic List

The counting has a clean repeating rule: turn n contributes exactly n numbers. Spot that pattern, add the turn sizes to see how many numbers have been said by the end of each turn, then find which turn the 53rd number falls in and read off its value inside that turn.

Execute — Answer: E

#5 Look for a Pattern 4.OA.C.5 Step 1

Count numbers per turn

On turn 1 Jo says 1 number.
On turn 2 Blair says 2 numbers.
On turn 3 Jo says 3 numbers.
Each turn the person says the numbers 1 up to the turn number, so turn n contributes exactly n spoken numbers.

$$\text{turn } n \text{ says } n \text{ numbers: } 1, 2, \ldots, n$$

💡 The turn number is the same as how many numbers get said that turn.

#7 Identify Subproblems 4.NBT.B.4 Step 2

Add up to find the turn

The total numbers said after turn k is 1 + 2 + ...
+ k.
Add turn by turn: after turn 9 the total is 1+2+3+4+5+6+7+8+9 = 45, and after turn 10 it is 45 + 10 = 55.
Since 53 is more than 45 but at most 55, the 53rd number is spoken during turn 10.

$$1+2+\cdots+9 = 45,\qquad 45 + 10 = 55,\qquad 45 < 53 \le 55$$

💡 Keep a running total of numbers said until it first reaches past 53.

#2 Make a Systematic List 4.OA.A.3 Step 3

Locate 53 inside turn 10

By the end of turn 9, 45 numbers have been said, so the 46th number is the first one of turn 10.
Turn 10 says 1, 2, 3, ..., 10 in order.
The 53rd number said is the 53 - 45 = 8th number of turn 10, and the 8th number in the list 1, 2, ..., 10 is 8.
So the 53rd number said is 8, which is answer (E).

$$53 - 45 = 8 \;\Rightarrow\; \text{8th number of } 1,2,\ldots,10 = 8$$

💡 Subtract the numbers already counted, then step that far into the current turn's list.

[1] #5 4.OA.C.5 On turn 1 Jo says 1 number. On turn 2 Blair says 2 numbers. On turn 3 Jo says 3

[2] #7 4.NBT.B.4 The total numbers said after turn k is 1 + 2 + ... + k. Add turn by turn: after

[3] #2 4.OA.A.3 By the end of turn 9, 45 numbers have been said, so the 46th number is the first

Review

Reasonableness: The running totals 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 confirm that position 53 sits between the 45th and 55th number, inside turn 10 whose list is 1..10. Counting 46,47,...,53 lands on the 8th entry, which is 8, one of the answer choices.

Alternative: Write these totals as triangle numbers T_k = k(k+1)/2. T_9 = 45 and T_10 = 55, so the 53rd number is the (53 - 45) = 8th term of turn 10, giving 8. This matches the running-sum count.

CCSS standards used (min grade 4)

4.OA.C.5 Generate a number or shape pattern following a given rule (Seeing that turn n always says exactly n numbers, 1 through n.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Adding the turn sizes into running totals to reach past 53.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Subtracting 45 from 53 to find the 8th position inside turn 10 and reading off its value.)

⭐ Add up how many numbers each turn says until you pass the one you want, then count that many steps into the last turn's list.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Count numbers per turn

Add up to find the turn

Locate 53 inside turn 10

Review

CCSS standards used (min grade 4)

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