Numsprite數靈精Play free
中文

Maths olympiad question · P5–P6★★★★ Expert

Queues: olympiad question 9

An expert question on queues for P5–P6, with a step-by-step solution and the answer. Queue questions are already in the Hua Xia Cup's P1 and P2 first rounds, and the P4 papers add square formations.

The question

At the sports day opening, pupils form a hollow square.

Each side of the outer layer has 13 pupils.

There are 2 layers from the outside in, each a square, packed tight, with a square space empty in the middle.

Then all of them re-form as a solid rectangle (not a square), nobody left over and nobody added, with length and width as close as possible.

How many pupils are on the rectangle's longer side?

Show the hints (3)
  1. Hint 1: First find how many pupils there are: add the layers, or take the empty middle from the big square.
  2. Hint 2: The empty middle is a square of 13 − 2 × 2 = 9 a side. Total = 13 × 13 − 9 × 9.
  3. Hint 3: 88 pupils. Length × width = 88, as close as possible: start near the square root of 88 and look downwards for a factor.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.The empty middle is a square of 13 − 2 × 2 = 9 a side, so there are 13 × 13 − 9 × 9 = 169 − 81 = 88 pupils (layer by layer gives the same: 48 + 40 = 88).
  2. 2.A solid rectangle: length × width = 88. The closer they are, the closer the width is to the square root of 88 (9 × 9 = 81, 10 × 10 = 100).
  3. 3.Try downwards from 9 for a factor of 88 (no square): 9 doesn't divide it; 88 ÷ 8 = 11 ✓.
  4. 4.So the rectangle is 11 × 8: 11 pupils on the longer side.
  5. ⚠️Common mistake: taking the hollow square as 13 × 13, or every layer as 4 × 13 (corners counted twice). And search for factors from near the square root, so length and width come out closest.

Answer: 11 pupils

Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download