August 2014
Problem of the Month

Special Numbers Problem



Given the following table of equations, write out the line that begins with 3773 using the same pattern.

13 + 23 - 13 = 8
23 + 33 - 13 = 34
33 + 53 - 23 = 144
53 + 83 - 33 = 610
83 + 133 - 53 = 2584
:
:
:
3773 + _____ - _____ = _____

What is the significance of these numbers?



Solution to the Problem:

3773 + 6103 - 2333 = 267914296

These are Fibonacci numbers, and the pattern is:

(2nd Fibonacci #)3 + (3rd Fibonacci #)3 - (1st Fibonacci #)3 = 6th Fibonacci #

(3rd Fibonacci #)3 + (4th Fibonacci #)3 - (2nd Fibonacci #)3 = 9th Fibonacci #

(4th Fibonacci #)3 + (5th Fibonacci #)3 - (3rd Fibonacci #)3 = 12th Fibonacci #

(5th Fibonacci #)3 + (6th Fibonacci #)3 - (4th Fibonacci #)3 = 15th Fibonacci #

(6th Fibonacci #)3 + (7th Fibonacci #)3 - (5th Fibonacci #)3 = 18th Fibonacci #

:
:
:
(14th Fibonacci #)3 + (15th Fibonacci #)3 - (13th Fibonacci #)3 = 42nd Fibonacci #

Did you notice that in the last line
14 + 15 + 13 = 42
and that it works for every line?

In general,
(n+1st Fibonacci #)3 + (n+2nd Fibonacci #)3 - (nth Fibonacci #)3 =
(n + (n+1) + (n+2)) Fibonacci #




Correctly solved by:

1. Kate Cotter Elmhurst, Illinois
2. Mike Bova Wallingford Connecticut
3. James Alarie Flint, Michigan
4. krishnanrk2000@yahoo.com ----------
5. Keith Mealy Cincinnati, Ohio
6. Elisa Arostegui Carmichael, California
7. Chad Fore Gate City, Virginia


Send any comments or questions to: David Pleacher