$1 + 8 = 9$

$2 + 9 = 11$

$3 + 10 = 13$

$4 + 11 = 15$

$5 + 12 = 17$

$6 + 13 = 19$

$7 + 14 = 21$

$8 + 15 = 23$

Alice from Perse School for Girls spotted a pattern:

There is a pattern, it's that the two numbers diagonal to each
other added together make the same number as the other pair of
diagonal numbers make.

Chris found another pattern using multiplication:

If you cross multiply the set of four numbers, their
difference is always $7$.

$1 \times 9 = 9$ and $2 \times 8 = 16$, difference $= 7$

$3 \times 11 = 33$ and $4 \times 10 = 40$, difference $=
7$

$15 \times 23 = 345$ and $16 \times 22 = 352$, difference $=
7$ etc...

If you discovered any other patterns, then do let us know. Please don't worry that your solution is not "complete" - we'd like to hear about anything you have tried.