Problem
Find cells from a window-sum rule
A 10×10 chart lists 1 to 100, ten per row. The four shapes may not be turned or flipped. Every covered cell must lie inside the chart. Place each shape so its cells total 96, 420, 123 and 332.
Reasoning · Grade 3-2 Conditions and Division
Your answer
Watch out Turning a shape to fit is tempting, but they may not be turned or flipped — only slid.
How to solve
Strategy Convert to Algebra — For each shape I write every covered cell as an offset from one anchor cell (right = +1, down = +10). The target sum becomes a simple equation in the anchor value; solving it gives the anchor, and the offsets give the rest. I then check the cells fit on the chart and re-add to confirm. The same equation also tells me when a target is impossible: if it forces the anchor to be a non-whole number, no placement exists.
1STEP 1
Place the staircase (sum 96)
The stair is a, a+11, a+22, so 3a + 33 = 96 gives 21, 32, 43.
a + (a+11) + (a+22) = 3a + 33 = 96 → a = 21
Each diagonal step adds the same 11, so the three numbers are evenly spaced and their total is just three times the middle number 32.
3.NBT.A.2Convert To Algebra2STEP 2
Place the plus/cross (sum 420)
The cross is five times its centre, so 420 ÷ 5 gives centre 84.
c + (c-1) + (c+1) + (c-10) + (c+10) = 5c = 420 → c = 84
The four arms cancel in pairs around the center, so the whole cross is just five copies of the center number.
3.OA.C.7Convert To AlgebraThe four arms of the cross cancel in pairs around the centre, so the whole cross is five copies of the centre number.
Why?
The arm above sits as far above the centre as the arm below sits beneath it, and the same holds left and right.
🧱The mean is the total shared out equallyThe average is the total divided by how many there are, so the total is the average times the count — and everything above that level gives away exactly what everything below it needs.
Why?
What one arm is above the centre the opposite arm is below, so the two changes cancel and the total does not move.
🧱What one part gains the other losesMove some amount from one addend to the other and the total does not budge, so you may make one number round for free.
3STEP 3
Place the hook (sum 123)
The hook gives 5a + 33 = 123, starting at 18.
a + (a+1) + (a+2) + (a+10) + (a+20) = 5a + 33 = 123 → 5a = 90 → a = 18
The three top cells average to the middle one (a+1), and the two cells below add another column-step of +10 and +20, so the whole hook reduces to 5 times the corner plus a fixed 33.
3.OA.A.4Convert To Algebra4STEP 4
Place the S (sum 332)
The S gives 4a + 20 = 332, starting at 78.
a + (a+1) + (a+9) + (a+10) = 4a + 20 = 332 → a = 78
The bottom pair sits one row down (+10) and one column left (-1), so it equals a+9 and a+10, keeping the equation simple.
3.NBT.A.2Convert To AlgebraAnswer
96: 21, 32, 43 / 420: 74, 83, 84, 85, 94 / 123: 18, 19, 20, 28, 38 / 332: 78, 79, 87, 88
Re-adding every placement confirms the targets: 21+32+43 = 96; 74+83+84+85+94 = 420; 18+19+20+28+38 = 123; 78+79+87+88 = 332. Each shape is kept upright (no rotation or flip) and every cell lies inside 1-100.
Takeaway
Pick one corner cell as your 'anchor,' write every other covered cell as +1 (right) or +10 (down) from it, and the target sum turns into one easy number sentence that hands you the anchor!
- Place the staircase (sum 96)
- Place the plus/cross (sum 420)
- Place the hook (sum 123)
- Place the S (sum 332)