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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

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