Back to school you lot...
Bletchley was crammed with top chess players.
A lot of people at Bletchley were top crossword experts too with a fair smattering of polyglots.
For a quick maths lesson:-
This is the 'original sequence': 12 26 46 72 104 142
The 'first differences' are, funnily enough, the differences between successive terms in the 'original sequence': 14,20,26,32,38
The 'second differences' are, funnily enough, the differences between the terms of the 'first differences': 6,6,6,6
To build a general formula we observe:
x
1=12
x
2=12+8+6
x
3=12+(8+6)+(8+6+6)
x
4=12+(8+6)+(8+6+6)+(8+6+6+6)
So, for x
n we have 12 plus 8*(n-1) plus a number of 6's that correspond to the 'triangular numbers' (1,1+2,1+2+3,1+2+3+4,1+2+3+4+5,... == 1,3,6,10,15,...) which we all know the formula for that. Think of the triangle like this:-
6
66
666
6666
To calculate the number of 6's we imagine a similar triangle spliced on to it to make a rectangle:-
6xxxx
66xxx
666xx
6666x
This makes a rectangle of size 4 * 5, of which the 6's make up only half. So a triangle of size has 4*(4+1)/2 elements or, in the general case: n*(n+1)/2
In this case the input is n-1, (with n=1 we have no 6's) so the formula for the number of 6's in x
n is (n-1)*(n-1+1)/2 = n(n-1)/2
So: x
n = 12 + 8*(n-1) + 6*n*(n-1)/2
= 12 + 8*(n-1) + 3*n*(n-1)
= 12 + 8n - 8 + 3n
2 - 3n
= 3n
2 + 5n + 4