Ken's Puzzle of the Week

Bob's New House

Bob: I'm depressed.  I moved here because the house number is special.  The house numbers run consecutively from 1, and when you add up all the house numbers on my street, the sum is exactly the square of my house number.  But a construction company is tearing down all the houses at one end of the street to build stores, and a second company is building more houses at the other end!

Sally: Don't worry, Bob.  Your house number will be just as special when the construction is complete.

What is Bob's house number and how many houses are on the street?

Source: Original.

Solutions were received from Nyles Heise, Mark Rickert, David Madfes, Kirk Bresniker, Denis Borris, Alex Ang, Joseph DeVincentis, and Jean Moreau de Saint-Martin.  I was only looking for the lowest house number, but I intentionally left it open to see what other solutions would be found.  Denis Borris reports these results:

I get multiple solutions:
b = Bob's number
n = last present street number
x = 1st street number after construction (has to be at low end)
m = last number after new homes added
x < b < n < m
b^2 = n(n + 1) / 2
m = {sqrt[1 - 4(x - x^2 - 2b^2)] - 1} / 2
Keeping n < 10,000 I get 21 solutions:
     x          b        n         m
     18         35       49        52
    196        204      288       348
    595       1189     1681      1783
    578       6930     9800      9817
    755       6930     9800      9829
   6194       6930     9800     11593
There are 18 with b=6930 and n=9800 :  21 cases for n < 10,000

Mail to Ken