# Sunday Times Teaser 2681 – Inconsequential

### by Victor Bryant

A sequence of integers in which the difference between consecutive terms is constant is known as an arithmetic sequence.  For example, 10, 29, 48, … is an arithmetic sequence with a ninth term (162) that is divisible by 9.

I have formed another arithmetic sequence that starts with two three digit numbers and has a ninth term that is also divisible by 9.  When I consistently replace different digits with different letters, this sequence starts ONE, TWO, … and its ninth term is NINE.

What nunber does WIN represent?

Set ONE = abc, TWO = dea = abc + k and NINE = bfbc = abc + 8k where a .. f are the six
different digits involved and k is the difference between terms. We then have:

NINE = 1000b + 100f + 10b + c = ONE + 8k = 100a + 10b + c + 8k

giving k = 125b + (25/2)(f – a) with f – a even. Hence we have:

b = 1 -> k = 125 +/- (50,100)
b = 2 -> k= 250 +/- (25, 75)
b = 3 -> k = 375 +/- (50,100)

From here we can reach the values of b, k, ONE, TWO, NINE and hence WIN by trial and error.