BUY BOOKS
Grid Sum Puzzle

Grid Sum

Grid Sum gives you a blank 3×3 (or larger) grid of algebraic expressions and target row and column totals; your task is to solve for x and place each expression so that every row and column sums correctly. It combines equation solving with logical placement.

Other Puzzle Types

Examples of Grid Sum

Grid Sum Example

Solve Grid Sum Puzzle

1. Compute the Grand Total of All Columns
  • Add together every column‐target total.
  • Call this value T.
  • For example, in a 3×3 grid with column targets 15, 18 and 21, you get T= 15 + 18 + 21 = 54.

2. Sum All the Algebraic Formulas Symbolically
  • Write down each of the grid’s nine expressions in terms of x, then add them up into one combined formula.
  • If your cells are labelled x+2, 2x−1, 3x, … then form F(x) = (x + 2) + (2x − 1) + (3x) + …

3. Set the Column Total Equal to the Formula Sum
  • Form the equation F(x) = T.
  • Solve for x, choosing the positive integer solution that fits the puzzle context.
  • Example: F(x) = (x + 2) + (2x − 1) + (3x) = 6x + 1, set 6x + 1 = 54 ⇒ 6x = 53 ⇒ x = 53/6 (reject, not an integer) or possibly different expression that yields an integer.

4. Substitute x into Every Formula
  • Compute each cell’s numeric value by replacing x with your solved number.
  • List the resulting nine (or 16, etc.) numbers beside the grid for easy reference.

5. Place Numbers by Process of Elimination
  • Start with any row or column whose target can only be made by one combination of the available numbers.
  • Write that number into the grid, then cross it off your reference list.

6. Iterate & Update Remaining Totals
  • After each placement, subtract the placed number from its row’s and column’s remaining target.
  • Use these reduced targets to find the next forced placement.

7. Final Validation
  • When all cells are filled, sum each row and column to confirm they match their original targets.
  • If they do, the puzzle is solved correctly.

Pro Tips
  • Check solvability early. If your solved x isn’t a whole positive number, re-check your symbolic sum or column‐total addition.
  • Work methodically. Ticking off each used number prevents accidental reuse.
  • Use mental estimates. Quickly scan small/large numbers – if a leftover target is 3 but your smallest unused number is 5, you know something’s off.
  • Keep neat notes. Write your symbolic sum in one place, the numeric substitutions in another, then your placement work in the grid itself.

By front‐loading the algebraic work (steps 1–4), you turn every Grid Sum into a straightforward logical placement exercise—no more rewriting each formula in turn!

Example Grid Sum Puzzle

Available expressions:
1. x
2. x + 1
3. x + 2
4. x – 1
5. 2x
6. 3x
7. x²
8. x² – 1
9. x² + 1

Initial table:
18 21 17
15 x² x – 1
19 x² – 1
22 x

Solution:
Sum of all the columns = 18 + 21 + 17 = 56 therefore T = 56
Sum of all the available expressions = x + x + 1 + x + 2 + x – 1 + 2x + 3x + x² + x² – 1 + x² + 1 = 3x² + 9x + 2 therefore f(x) = 3x² + 9x + 2
F(x) = T therefore 3x² + 9x + 2 = 56
3x² + 9x – 54 = 0
Simplified this is x² + 3x – 18 = 0
Factorised this is (x – 3)(x + 6)
So x = 3 and x = -6.
We use the positive value of x so x = 3

Next we work out the values of each of the available expressions.
1. x = 3
2. x + 1 = 4
3. x + 2 = 5
4. x – 1 = 2
5. 2x = 6
6. 3x = 9
7. x² = 9
8. x² – 1 = 8
9. x² + 1 = 10

And now we put the formulas back in the grid, in the correct places:
18 21 17
15 x² -> 9 x + 1 -> 4 x – 1 -> 2
19 2x -> 6 x² – 1 -> 8 x + 2 -> 5
22 x -> 3 3x -> 9 x² + 1 -> 10
By balancing arithmetic and spatial reasoning, Grid Sum hones your ability to juggle multiple equations at once and reinforces systematic checking of both rows and columns – great practice for multi-constraint problems.

Now lets practice ...

We have added a selection of our puzzles so that you can practice online: Practice Now