Label this plum tree graph to make it totally magic!
Find all the ways of placing the numbers 1 to 9 on a W shape, with
3 numbers on each leg, so that each set of 3 numbers has the same
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
That's . . . . . 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17
Added to . . 17 + 16 + 15 + 14 + 13 + 12 + 11 + 10
The pairs 10 with 17, 11 with 16, and so on to 17 with 10, all
make a sum of 27, and there are 8 pairs in all doing that.
Using this same method to find the sum
of : 10, 11, 12, . . . . . . . . . 79, 80?
71 pairs each with a sum of 90, makes 6390, and half of that is
The first number is s so
the last number is s + n -
The pairs all have a sum of 2s
+ n - 1 and there are n pairs like that.
Finally the sum we want is half of that, which in algebra is . .
. . n (2s + n -1) / 2
16 and 32 are pure powers of 2 : they are $2^4 and 2^5$
We are looking for a run of consecutive numbers that totals to
Whether s is an odd or an even number, 2s will be even , so n - 1 must also be even , because the
whole factor (2s + n - 1) has to be some power of two.
When n -1 is even, n itself
will be odd , but 16 does not have any odd factors, so there
is no value, odd or even, from which to start a run of consecutive
numbers whose sum will be 16.
We can also see that the same reasoning would apply to making 32
and any other pure power of 2 .
Try this reasoning, based on odd-ness
and even-ness, while looking at the rectangle of dots on the main
Do you prefer to reason from the algebra
or from the image, or do you find that taking the algebra and the
image together is somehow the best way to increase your confidence
in the validity of your argument ?