Reasoning · Grade 3-2 Conditions and Division

Problem

Capstone: grid-sum and remainder conditions

On a chart with seven numbers per row, take a 3×3 frame. All nine cells must hold numbers and stay inside the chart. The sample frame totals 81. Find the largest number in the frame that totals 279.
Your answer
How to solve
Strategy Look for a Pattern — I use the sample frame (sum 81) to discover the rule that the nine numbers in a 3x3 frame are symmetric around the center, so their sum equals 9 times the center. That rule turns the sum 279 directly into the center number, and the chart's +1 right / +7 down structure gives the largest cell from the center.
1STEP 1

Find the rule from the sample frame

Pairing opposite cells makes the total nine times the centre.

1+17 = 3+15 = 8+10 = 2+16 = 18 = 2 × 9, 9 × 9 = 81
2STEP 2

Use the rule to find the center of the second frame

279 ÷ 9 makes the centre 31.

279 ÷ 9 = 31
3STEP 3

Locate the largest cell from the center

The bottom-right corner is 8 past the centre: 39.

31 + 7 + 1 = 39
Answer
39
279 ÷ 9 = 31, 31 + 8 = 39
The second frame is 23,24,25 / 30,31,32 / 37,38,39; their sum is 9 x 31 = 279, matching the clue, and 39 is indeed the largest. The largest cell (39) is plausibly bigger than the sample frame's largest (17), consistent with a larger total.
Takeaway

A 3x3 frame's numbers balance around the middle, so the total is just 9 times the center - and the biggest number sits in the bottom-right corner!

  • Find the rule from the sample frame
  • Use the rule to find the center of the second frame
  • Locate the largest cell from the center