How Old Am I?

This one produced a bumper crop! Well done the following who touched first base: KS3 Maths Club, Strabane Grammar School; Lucinda, Stamford High School, Jack, Hethersett High School; Claire, Madras College; Arwa, The International School, Brunei and Jonathan & Charlie from St Peter's College, Adelaide, Austral ia.

Kang of The Chinese High School of Singapore used algebra as follows:

Let Reza's present age be $x$

$x+3 = (x-3)^2$

$\Rightarrow x+3 = x^2 - 6x+9$

$\Rightarrow x^2-7x+6=0$

$\Rightarrow (x-1)(x-6)=0$

Therefore either $x = 1$ or $6$

Since Reza is at least $3$ years old, he must be $6$.

Renate and Amy of Hethersett High School suggested a very good method (see the table below). They noted that you can start with column (iii), then fill column (iv) with the squares of the numbers in (iii), then the age in column (i) is the mean of columns (iii) and (iv).

Similar tables came from Y8, Y9 and Y10 The Mount School York where they noticed that the ages are triangle numbers, as did Ying of Tao Nan School, Singapore, and also Mark ofKing Edward VI High School, Birmingham and Steven ofBedlington High School, Nothumberland.

(i)	(ii)	(iii)	(iv)
Age	$k$	Age $k$ years ago	Age $k$ years ahead
$1$	$0$	$1$	$1$
$3$	$1$	$2$	$4$
$6$	$3$	$3$	$9$
$10$	$6$	$4$	$16$
$15$	$10$	$5$	$25$
$21$	$15$	$6$	$36$
$28$	$21$	$7$	$49$
$36$	$28$	$8$	$64$
$45$	$36$	$9$	$81$
$55$	$45$	$10$	$100$

If you look at the first two columns, another pretty pattern emerges from the fact that every square number splits into two consecutive triangle numbers. Try investigating this with dot patterns and algebra.