Number sequences present all sorts of intriguing puzzles and patterns.
Consider, for example, the sequence of counting numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 . . . .
Now, take out every second number, leaving: 1 3 5 7 9 11 13 15 . . . ; form the cumulative totals of these numbers: 1 (1 + 3) (4 + 5) (9 + 7) (16 + 9) (25 + 11) (36 + 13) (49 + 15) . . . ; and out pops the sequence of consecutive squares: 1 4 9 16 25 36 49 64 . . . !
This seemingly magical transformation of one sequence into another was first discovered and explored by mathematician Alfred Moessner in the early 1950s. He and others found a host of such relationships between different number sequences.
Again, starting with the sequence of counting numbers, suppose you take out every third number, add up what's left to get cumulative totals, then remove every second number in the new list and total the remaining numbers. What do you end up with?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 . . .
1 2 4 5 7 8 10 11 13 14 . . .
1 3 7 12 19 27 37 48 61 75 . . .
1 7 19 37 61 . . .
1 8 27 64 125 . . .
The sequence of cubes!
If you go through the same procedure again, this time striking out every fourth number at the start, the result should come as no surprise. You end up with the sequence of fourth powers: 1 16 81 256 . . . . In general, taking out the nth number to start with gives a sequence of nth powers in the end.
What happens if you take out the so-called triangular numbers: 1 (1 + 2) (1 + 2 + 3) (1 + 2 + 3 + 4) . . . (1 + 2 + 3 + 4 + . . . n) and, as before, calculate cumulative totals, then take out the appropriate numbers from the new list, and so on, as above?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 . . .
2 4 5 7 8 9 11 12 13 14 . . .
2 6 11 18 26 35 46 58 71 85 . . .
6 18 26 46 58 71 . . .
6 24 50 96 154 225 . . .
24 96 154 . . .
24 120 274 . . .
120 . . . .
Notice that the numbers down the left side, 1 2 6 24 120, are the factorial numbers: 1 (1 x 2) (1 x 2 x 3) (1 x 2 x 3 x 4) (1 x 2 x 3 x 4 x 5) or, in general, (1 x 2 x 3 x … x n). Somehow, the recipe turns addition into multiplication.
I first came across these surprising sequence transformations when Richard Guy described them at a meeting on recreational mathematics held in 1986 at the University of Calgary. Now this material -- and much, much more -- is included in a fascinating new book by Guy and John Conway called The Book of Numbers. If you want to stretch your mind from the integers to the surreals, this is the book to read!
Copyright © 1996 by Ivars Peterson.
Conway, J.H., and R.K. Guy. 1996. The Book of Numbers. New York: Copernicus.
Guy, R.K., and R.E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History. Washington, D.C.: Mathematical Association of America.
Peterson, I. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York, W.H. Freeman.
Sloane, N.J.A., and S. Plouffe. 1995. The Encyclopedia of Integer Sequences. New York: Academic Press.
Comments are welcome. Please send messages to Ivars Peterson at ip@scisvc.org.
Ivars Peterson is the mathematics and physics writer at Science News. He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, and *Fatal Defect (now available in paperback). His current work in progress is The Jungles of Randomness (to be published in 1997 by Wiley).
*NOW AVAILABLE in paperback: Fatal Defect: Chasing Killer Computer Bugs by Ivars Peterson. Vintage Books, 1996. ISBN 0-679-74027-9. U.S. $13.00.
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