Suppose we have the N vertices of a regular N-gon inscribed in a circle of radius 1. Select one vertex W and draw line segments from W to each of the other N-1 vertices. What is the total product of the lengths of these line segments?
(old problem - original author unknown)
Each line segment is a chord of the circumscribing circle, subtending 360*i/n degrees, where i goes from 1 to n-1. The length of such a chord is twice the sine of half the subtended arc. So, using Prod to signify what is usually denoted by a capital Pi, the formula would be Prod{i=1 to n-1} 2 sin(180 i / n deg). Tabulated for n = 3 to 25, this comes out to
3 3.00000000000000000000000000000000000000000000000000000000000001
4 4.00000000000000000000000000000000000000000000000000000000000007
5 5.00000000000000000000000000000000000000000000000000000000000008
6 6.0000000000000000000000000000000000000000000000000000000000002
7 7.00000000000000000000000000000000000000000000000000000000000063
8 8.00000000000000000000000000000000000000000000000000000000000064
9 9.00000000000000000000000000000000000000000000000000000000000084
10 10.0000000000000000000000000000000000000000000000000000000000008
11 11.00000000000000000000000000000000000000000000000000000000000158
12 12.00000000000000000000000000000000000000000000000000000000000224
13 13.00000000000000000000000000000000000000000000000000000000000275
14 14.00000000000000000000000000000000000000000000000000000000000305
15 15.00000000000000000000000000000000000000000000000000000000000361
16 16.0000000000000000000000000000000000000000000000000000000000048
17 17.00000000000000000000000000000000000000000000000000000000000476
18 18.00000000000000000000000000000000000000000000000000000000000467
19 19.00000000000000000000000000000000000000000000000000000000000578
20 20.00000000000000000000000000000000000000000000000000000000000718
21 21.0000000000000000000000000000000000000000000000000000000000102
22 22.00000000000000000000000000000000000000000000000000000000000912
23 23.00000000000000000000000000000000000000000000000000000000000929
24 24.00000000000000000000000000000000000000000000000000000000001086
25 25.00000000000000000000000000000000000000000000000000000000001551
indicating that the non-zeros near the ends of these numbers are the result of rounding errors, and that the simplified formula for the product is actually N.
For N=3, for example, the sides of the triangle are sqrt(3). Only two of the sides are multiplied together, so the result is 3. For a square (N=4), the sides are sqrt(2) and the diagonal is 2. The product of two sides and the diagonal is 4.
4 point 13
5 Pi=atan(1)*4:Dr=Pi/180
10 for N=3 to 25
20 P=1
30 for I=1 to N-1
40 P=P*2*sin(180*I*Dr/N)
50 next I
60 print N,P
70 next N
Edited on October 20, 2005, 2:09 pm
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Posted by Charlie
on 2005-10-20 13:54:01 |
No proof, but ... (spoiler)
