Number Curiosities – Part One: First Powers.

Uncle Bob

First powers of numbers are just the numbers themselves, so why did I choose this curious subtitle? Answer: I’m setting you up for curiosities among the second and third powers in future articles.

The Nearness of Them
Use the digits {1, 2, 3, 4, 5, 6} to make two numbers closest to one another. We try 345 and 261 with a difference of 84. A better result comes from 165 and 234 – difference 69. Is there a closer pair using these digits? Yes. The closest pair have a difference of 47. Find it.

Find the closest pairs using {0, 1, 2, 3, 4, 5, 6, 7} and then find the smallest difference between a pandigitial pair – a pair that uses all digits zero thru 9 once. You will find the number 47 involved twice more. Very curious.

No Collisions?
We construct a partition of 15 into triplets with certain common differences. Example: the triplet {2, 4, 6} has a common difference of 2.

We start with (1, 8, 15} with a common difference of 7; we look for four more triplets with differences of 5, 4, 2, and one. We find {2, 7, 12}, {6, 10, 14}, {9, 11, 13}, and {3, 4, 5}. Even though we used five different common differences, we have covered all 15 and managed to avoid repeating a number – a collision.

Can you find a different partition of these fifteen numbers into triplets that have these same common differences? The triplet {1, 8, 15} is still required. I found my solution with a curious method.

Sums, More and Fewer
Consecutive whole numbers have a common difference of one; for instance (19, 20, 21, 22, 23}. 105 happens to be the sum of those numbers. 105 can also be expressed as 34 + 35 + 36, another series of consecutive whole numbers. Find 5 other ways 105 is the sum of consecutive numbers.

Find all the ways 120 is the sum of consecutive whole numbers. Curiously there won’t be as many. What might determine how many ways?

Sums of consecutive whole numbers are never a power of 2. Powers of 2 have no odd factors. The sums can be odd or even but never among the set {2, 4, 8, 16, 32, and so on}. You can assist me with a proof. For brevity’s sake we’ll refer to an “odd list” as an odd number of consecutive whole numbers, like {19, 20, 21, 22, 23}; and an even list as one with an even count, for example {5, 6, 7, 8}.

The proof has two parts: one for an odd list and one for an even list.

The average of an even list is not a whole number. {5, 6, 7, 8} has a mean of 6.5. The sum is 6.5 times 4, or 26. It’s an even sum but not a power of two. It has a factor 13 because 6.5 is thirteen halves. The mean of any even list will be an odd number of halves (why?) and have an odd factor in its sum.

Can you establish the proof for an odd list? Hints: The set {19, 20, 21, 22, 23} has a mean of 21 and a sum of 105 – obviously not a power of 2. But what about the sets {10, 11, 12, 13, 14} or one that’s centered on a power of two as in {15, 16, 17}. Find the means and sums of these sets and then draw a general conclusion.

The Shape of Things
A number that is the sum of a set {1, 2, 3, …, n} is called triangular. We can illustrate the reason for this name by diagramming {1, 2, 3, 4}.

@
@@
@@@
@@@@

So the sum ten is a triangular number as are 105 and 120. Those two are the sums of 1 through 14 and 1 through 15 respectively. The sum of 105 and 120 is 225, a square number and a second power of 15. We’ll look for curiosities among the second powers next month.

solutions