Question

There are 2014 squares in the diagram, with number 7 in the first square and 6 in the 9th square. The sum of the numbers in any three consecutive squares is 21. Find the number in 2015th square.

Answer 1

7 8 6 7 8 6 7 8 6 .....

2015 mod 3 = 2
The second digit in 7,8,6 is 8.

The 2015th number is 8.

Answer 2

I can't understand your method. Please describe the sum in detail.

Answer 3

"The sum of the numbers in ANY three consecutive squares is 21."
implies the first three numbers are repeated over and over again. Do you agree with that?
x y z
If x+y+z = 21, then the 4th number must be an x: x y z x because now y+z+x = 21. Then the 5th number must be a y: x y z x y because now z+x+y = 21.

The 1st, 4th, 7th, 10th, 13th, ... numbers are identical and they are all 7.
The 2nd, 5th, 8th, 11th, 14th, .... numbers are identical.
The 3rd, 6th, 9th, 12th, 15th, ... numbers are identical and they are all 6.

From the above we can now conclude:
The 2nd, 5th, 8th, 11th, 14th, .... numbers are identical and they are all 8 because 7 + 8 + 6 = 21.

So the sequence of numbers are: 7 8 6 7 8 6 7 8 6 .....
How many full groups of three are there in 2015?
2015 / 3 = 671 + 2/3
So there will be 671 full groups of 7 8 6.
671 * 3 = 2013. 2015 - 2013 = 2. So the 2014th term is 7, the 2015th term is 8.