Thread: the House Problem

Consider this series: 1, 2, 3,..., N-1, N, N+1,..., M.
Given 1 + 2 + 3 + ... + (N-1) = (N+1) + (N+2) +...+ M. Find N.
(hint: N is a 4 digits number)

Originally Posted by john_n82

Consider this series: 1, 2, 3,..., N-1, N, N+1,..., M.
Given 1 + 2 + 3 + ... + (N-1) = (N+1) + (N+2) +...+ M. Find N.
(hint: N is a 4 digits number)

Note,

Therefore, , therefore .
We need the discriminant to be a square, so we need .
This is a Pellian equation.

Originally Posted by ThePerfectHacker

Note,

Therefore, , therefore .
We need the discriminant to be a square, so we need .
This is a Pellian equation.

I found M = 8, so N = 6.

Originally Posted by john_n82

Consider this series: 1, 2, 3,..., N-1, N, N+1,..., M.
Given 1 + 2 + 3 + ... + (N-1) = (N+1) + (N+2) +...+ M. Find N.
(hint: N is a 4 digits number)

i think you mean N^2 is a four-digit number

in this case M = 49, N = 35

i found 2 more values for M

M = 288, N = 204
M 9800, N = 6930

What is the exact answer for this problem?

Originally Posted by john_n82

i found 2 more values for M

M = 288, N = 204
M 9800, N = 6930

What is the exact answer for this problem?

There are many answers to this problem.
You are told that you need N to be a 4 digit number so N=6930