Algebra Play

Use letters to explain number tricks and patterns: the 1089 trick, reversed digits, calendar squares and the nth term of a pattern.

Key ideas

Write the unknown number as a letter and follow every step of the trick; what is left shows why it works.
A two-digit number with digits a and b is 10a + b, not ab.
The 1089 trick: take a three-digit number whose first and last digits differ by at least 2, subtract its reverse, then add the result to its own reverse: the answer is always 1089.
In a 3 × 3 square of calendar dates with middle date m, the dates are m ± 1, m ± 6, m ± 7 and m ± 8.
A pattern that grows by the same amount each step has nth term an + b.

Two-digit number	10a + b
Number and its reverse	(10a + b) + (10b + a) = 11(a + b); difference = 9(a − b)
Calendar 3 × 3 square	sum of all nine = 9m; each diagonal = 3m
Growing pattern	nth term = (step) × n + (first term − step)

In a 3 × 3 square of dates on a calendar the middle date is 17. Find the sum of each diagonal and explain why they are equal.

Working: (m − 8) + m + (m + 8) = 3m and (m − 6) + m + (m + 6) = 3m

Answer: 51 each