Algebra grids and shape puzzles

You have seen puzzles where a square, a circle, and a triangle stand for three secret numbers, and the row totals (or column totals) are given. The challenge: find what each shape is worth. These look like games, but they are really systems of linear equations , every row is one equation, every shape is one unknown.

Concept

Reading a grid. A typical algebra grid looks like this:

Each row says "the sum of these shape-numbers equals the given total."

Setting up equations. Let each shape stand for an unknown letter:

• Let $\square = s$, $\circ = c$.
• Row 1: $s + s + s = 27$, i.e., $3s = 27$, so $s = 9$.
• Row 2: $c + c + s = 19$, i.e., $2c + 9 = 19$, so $2c = 10$ and $c = 5$.

The grid is solved: $\square = 9$, $\circ = 5$. The trick was to find a one-shape row first (here Row $1$, all squares), solve it directly, then substitute into the next row.

When no row has all one shape. Combine rows. If Row 1 says $\square + \circ = 10$ and Row 2 says $\square + 2\circ = 14$, then subtracting Row 1 from Row 2 gives $\circ = 4$, hence $\square = 6$.

Three shapes, three rows. With three unknowns you need three independent equations. Look for the row with the fewest distinct shapes , that is the easiest to use first.

Bigger grids with column sums too. When both row and column totals are given, you have more equations than unknowns. Use the simplest equation first; the others act as a check. If a check fails, the puzzle is inconsistent , somewhere you (or the puzzle's author) made an arithmetic slip.

A general lesson. Real-life "shape" puzzles aren't just for fun. Replace shapes with quantities (apples, pencils, costs) and the same technique solves shopping bills, school-event budgets, and stock-keeping problems.

Worked examples

Example 1. Solve the grid:

• $2s + c = 27$ and $s + 2c = 21$.
• Add them: $3s + 3c = 48$ so $s + c = 16$.
• From row 1: $s + (s+c) = 27 \Rightarrow s + 16 = 27 \Rightarrow s = 11$.
• Then $c = 16 - 11 = 5$.
• Check row 2: $11 + 2 \cdot 5 = 21$. ✓

Example 2. Three shapes , $\square, \circ, \diamondsuit$ (let them be $s, c, d$). Rows: $s + s + c = 20$, $c + c + d = 16$, $d + d + s = 12$. Find each.

• $2s + c = 20$, $2c + d = 16$, $2d + s = 12$.

• Sum: $3s + 3c + 3d = 48 \Rightarrow s + c + d = 16$.

• From row 1: $c = 20 - 2s$. From row 3: $d = (12 - s)/2$.

• Substitute into $s + c + d = 16$: $s + (20 - 2s) + (12 - s)/2 = 16 \Rightarrow (4s/2) - 2s + (12 - s)/2 + 20 = 16$… simpler: $2s + 2(20 - 2s) + (12 - s) = 32 \Rightarrow 2s + 40 - 4s + 12 - s = 32 \Rightarrow -3s + 52 = 32 \Rightarrow s = \dfrac{20}{3}$.

  This indicates the puzzle would have non-integer values , fine in algebra, but a sign to double-check the numbers in real puzzles. The method however is correct.

Example 3. Two rows: $\bullet + \blacklozenge + \blacklozenge = 18$, $\blacklozenge + \bullet + \bullet = 15$. Find each.

• Let $\bullet = b, \blacklozenge = d$. Then $b + 2d = 18$ and $2b + d = 15$.
• Doubling the second: $4b + 2d = 30$. Subtract first: $3b = 12 \Rightarrow b = 4$, then $d = 7$.
• Check: $4 + 7 + 7 = 18$. ✓

Example 4. A magic-square style grid: row sums are all $15$ and the grid is $3 \times 3$ filled with shapes. If one row has $\square + \circ + \triangle = 15$ and the diagonal has $\square + \square + \square = 15$, find $\square$.

• $3\square = 15$ so $\square = 5$. (Then $\circ + \triangle = 10$.)

Try it yourself

1. $\square + \square = 14$. Find $\square$.
2. Two rows: $\square + \circ = 13$ and $\square - \circ = 3$. Find each.
3. Rows $3 \blacktriangle = 18$ and $2 \blacktriangle + \blacksquare = 16$. Find both.
4. $\square + \circ + \circ = 17$, $2 \square + \circ = 13$. Find each.
5. Three rows: $a + b + c = 6$, $2a + b = 7$, $a + 2c = 8$. (Use letters directly , no shapes.)
6. $\circ + \circ + \circ + \square = 22$ and $\circ + 3 \square = 18$. Find each.
7. In a row of $4$ identical squares totalling $36$, find each.
8. $\diamond + \diamond + \star = 14$ and $\star + \star + \diamond = 13$. Find each.

Activity

Make a shape puzzle. Pick two numbers , your favourite, say $7$ and $11$. Design two rows whose totals are determined by these numbers. For example: "$3 \square + \circ = 32$" and "$\square + 2 \circ = 25$". Hand the puzzle to a friend (without revealing $7$ and $11$) and challenge them to find the shape values. Then show them the algebra you used.