Bob's New House
Bob moved to a new house on another long street and noticed that now
the sum of the house numbers up to his own house, but excluding it,
equals the sum of the numbers from his house to the end of the road,
again excluding his house. If the houses are numbered consecutively,
starting from 1, what is the number of Bob's house?
Find the number of houses on the road, and Bob's house number if
his has 1, 2, 3, or 4 digits.
Source: Similar to a previous problem (3/26/97) from
The Penguin Book of Curious and Interesting Puzzles,
David Wells, 1992.
Solutions were received from:
The first few solutions for Bob's house are:
Nick Baxter showed that each successive answer can be found from the
previous ones with the following equations:
B(i)=6*B(i-1) - B(i-2)
L(i)=6*L(i-1) - L(i-2) + 2
Israel Eisenberg found two other equations only dependent upon one past
answer (they can be derived from those above):
B(n+1) = 3*B(n) + 2*L(n) + 1
L(n+1) = 4*B(n) + 3*L(n) + 1
Most solvers used an exhaustive computer search after finding a representative
equation between the two variables. Nick Baxter provides this analytical
C(B,2) = C(N+1,2) - C(B+1,2)
or 2B^2 = N(N+1)
Solving for N, N = (-1+sqrt(8n^2+1))/2
For N to be integral, 8n^2+1 must be a square.
Let x^2 = 8b^2+1, a Pell equation.
Sources for Pell Equations:
Slide Rule's Games,
Eric's Treasure Trove of Mathematics.
Mail to Ken