2026 AMC 8 Problem 2: Solution

Work through 2026 AMC 8 Problem 2 on counting and sums with a written solution, hints, and a visual explanation.

Number TheoryLevel 1 · Warmup

In the array shown below, three 3s are surrounded by 2s, which are in turn surrounded by a border of 1s. What is the sum of the numbers in the array?

A five-row, seven-column array. The outer border is all 1s; the next border is all 2s; the middle row has three 3s at its center.
Diagram 1
  • A. 49
  • B. 51
  • C. 53
  • D. 55
  • E. 57
Hints
  1. Break the array into its 5 rows and add each row separately.
  2. Notice the array is symmetric top-to-bottom — the top and bottom rows match, and so do the 2nd and 4th rows.
  3. Add the five row sums together for the total.
Read the step-by-step solution

Answer: C · 53

  1. Count each layer

    The array is 5 rows by 7 columns. Sum each row: Row 1 = 7, Row 2 = 1+2+2+2+2+2+1 = 12, Row 3 = 1+2+3+3+3+2+1 = 15.

  2. Use symmetry

    By symmetry, the total = 7 + 12 + 15 + 12 + 7 = 53.

    7 + 12 + 15 + 12 + 7 = 53

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