Calendar magic

A calendar looks like a plain grid, but the rule "go right by $1$, go down by $7$" hides quite a bit of algebra. With it you can build "magic" tricks: ask a friend to circle a $2 \times 2$ block of dates and tell you only their sum. You can recover all four dates. Or you can grab a $3 \times 3$ block , the central date is always one-ninth of the total. Algebra explains everything.

Concept

The calendar rule. If $a$ is any date in a month, then the date one column to the right is $a + 1$ (the next day) and the date one row below is $a + 7$ (the same weekday a week later). So a $2 \times 2$ block on a calendar looks like this:

$\begin{array}{|c|c|} \hline a & a+1 \\ \hline a + 7 & a + 8 \\ \hline \end{array}$

Sum of a $2 \times 2$ block. Add the four numbers:

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

So if the sum is $S$, then $a = \dfrac{S - 16}{4}$. From $a$ you read off the other three.

Example. Friend says "sum is $36$". Then $4a + 16 = 36 \Rightarrow 4a = 20 \Rightarrow a = 5$. The block is $\{5, 6, 12, 13\}$.

$3 \times 3$ block magic. Let the centre date be $c$. The block is:

$\begin{array}{|c|c|c|} \hline c-8 & c-7 & c-6 \\ \hline c-1 & c & c+1 \\ \hline c+6 & c+7 & c+8 \\ \hline \end{array}$

(Each row differs by $7$ from the next; each column differs by $1$ from the next.) Summing all nine:

$9c.$

The minus and plus terms cancel out. So the sum of any $3 \times 3$ calendar block is exactly $9$ times its centre. Friend says "sum is $171$"; then the centre is $171/9 = 19$, and you can name all nine dates.

Some other slick patterns.

• A horizontal $1 \times 4$ block (say $a, a+1, a+2, a+3$) sums to $4a + 6$.
• A vertical $4 \times 1$ block ($a, a+7, a+14, a+21$) sums to $4a + 42$.
• A "diagonal" pair $\{a, a+8\}$ and $\{a+1, a+7\}$ both sum to $2a + 8$ , a knight's-move identity.

These tricks work in any month on any calendar, because the relative arrangement of dates in a grid is fixed. You just need to be careful near the edges of the month, where the pattern can be cut short.

Worked examples

Example 1. $2 \times 2$ block from a calendar sums to $40$. Find the four dates.

• $4a + 16 = 40 \Rightarrow a = 6$. Block: $6, 7, 13, 14$.

Example 2. $2 \times 2$ block sums to $80$. Find the dates.

• $4a + 16 = 80 \Rightarrow a = 16$. Block: $16, 17, 23, 24$.

Example 3. $3 \times 3$ block on a calendar has sum $108$. Find the centre and the dates.

• Centre $= 108/9 = 12$. The block has $4, 5, 6, 11, 12, 13, 18, 19, 20$.

Example 4. A friend picks a row of three consecutive dates (e.g. $9, 10, 11$). Their sum is $S$. Find a formula for the middle date.

• Sum $= 3 \cdot (\text{middle})$, so middle $= S/3$. Quick and elegant.

Try it yourself

1. A $2 \times 2$ block on a calendar sums to $52$. Find the dates.
2. A $2 \times 2$ block sums to $96$. Find the dates.
3. A $3 \times 3$ block sums to $90$. Find the centre.
4. A horizontal block of $5$ consecutive dates has middle $20$. Find the sum.
5. In a $2 \times 2$ block, the bottom-right is $19$. Find the sum of the block.
6. A $4 \times 1$ vertical column has sum $70$. Find the top date.
7. Why does a $3 \times 3$ calendar block always sum to $9$ times its centre?
8. Invent a $2 \times 3$ block calendar trick. What is its sum in terms of the top-left $a$?

Activity

Calendar guess. Open your favourite month. Cover the calendar with your hand and ask a friend to draw a $2 \times 2$ box of dates and tell you only the sum. Use the formula $a = (S - 16)/4$ to find the top-left and announce all four dates correctly. Try it three times. Then explain why it always works , using algebra.