13.4 Fun with Grids

Calendar magic

Pick any 2\times 2 block from a month of the calendar. A friend adds its four numbers and tells you only the sum, yet you can name all four numbers. How?

Look at the layout. On a calendar the box to the right is one more and the box below is seven more. So if the top-left number is a, the block reads:

\begin{array}{cc} a & a+1 \\[2pt] a+7 & a+8 \end{array}

The four numbers add to

a + (a+1) + (a+7) + (a+8) = 4a + 16.

Worked Example

A friend’s 2\times 2 calendar block sums to 56. Find the four numbers.

4a + 16 = 56 \;\Longrightarrow\; 4a = 40 \;\Longrightarrow\; a = 10.

So the block is 10,\ 11,\ 17,\ 18. (Check: 10+11+17+18 = 56 ✓.)

Worked Example

Another friend’s 2\times 2 block sums to 76. Find the four numbers.

4a + 16 = 76 \;\Longrightarrow\; 4a = 60 \;\Longrightarrow\; a = 15.

So the block is 15,\ 16,\ 22,\ 23. (Check: 15+16+22+23 = 76 ✓.)

Math Talk Invent your own calendar trick. Try a different shape, a 2\times 3 block, an L-shape, or a 3\times 3 block. Write each cell in terms of the top-left a, add them up, and find the rule that pulls a back out of the sum.

Algebra grids

In these grids, shapes stand for numbers, and the last entry in each row is the sum of the values to its left. Reading the rows gives equations.

Suppose three identical squares in a row sum to 24, and a row of square, square, circle sums to 20:

\square + \square + \square = 24 \;\Longrightarrow\; 3\,\square = 24 \;\Longrightarrow\; \square = 8.

\square + \square + \bigcirc = 20 \;\Longrightarrow\; 8 + 8 + \bigcirc = 20 \;\Longrightarrow\; \bigcirc = 4.

(If instead the second row were \bigcirc + \bigcirc + \square = 20, then 2\,\bigcirc + 8 = 20, giving \bigcirc = 6.) Once each shape’s value is settled, every empty square can be filled in.

The Idea A grid puzzle is simply a tidy way to write a system of equations in pictures. Find a row that locks down one shape, feed its value into another row to fix the next, and carry on.

---