If you begin arranging stones in patterns, you quickly find associations between geometrical shapes and numbers. Some numbers can be arranged in triangles, for example:
There is even a mathematical significance to these numbers T(n) -- they are the sums of consecutive integers, starting with 1: 1, 1+2, 1+2+3, etc. By rearranging the stones slightly, and putting in another copy of the triangle, you can find an even simpler formula for T(n):
This gives a formula for 1+2+3+...+n which is sometimes handy to know, and which looks rather mysterious without the geometrical picture to remind you where it came from.
The perfect squares are a better known example of numbers associated with shapes:
There is even a neat connection between the square numbers and the triangular numbers, based on the geometrical fact that you can put two triangles together to make a square:
Another way to look at square numbers S(n) shows that they are the sums of consecutive odd integers: 1, 1+3, 1+3+5, ...
As you see above, the decomposition S(n)=T(n)+T(n-1) also leads to the sum of consecutive odd integers, since T(n) is the sum of consecutive integers.
These are good examples of two branches of mathematics, arithmetic and geometry, interacting in a not completely obvious way!
"The earliest precursor of writing," Denise Schmandt-Besserat, Scientific American, June 1978, p. 50.
The article describes Neolithic objects apparently
used as counters, and how their manipulation may have led to the
invention of writing.
A number is called "perfect" if it is the sum of its distinct proper factors (i.e., all the integer factors, including 1, but not including the number itself). For example 6=2x3, so the proper factors of 6 are (1,2,3), and 6=1+2+3. Another example is 28=2x2x7: the distinct proper factors are (1,2,4,7,14), and 28=1+2+4+7+14. Here is a sort of pattern:
6=(1+2)x2 is perfect.
28=(1+2+4)x2x2 is perfect.
120= (1+2+4+8)x2x2x2 is NOT perfect, but
496=(1+2+4+8+16)x2x2x2x2 IS perfect. See a
pattern? Does it really work?
Two integers are "friendly" if each is the sum of the proper factors of the other. For example, 220 is friendly -- I leave it to you to find the friend, and verify that it is a true friend.
Invent your own class of numbers, using geometrical arrangements of things, and point out some interesting property (-ies) of these numbers. Be sure your idea is different from everybody else's! (If it is inventive enough, you can be sure it will be different.)