Inspiring Icosioctagon

Geometry Level 5

7 cot ⁡ π 28 = n 1 sin ⁡ π 14 + n 2 sin ⁡ π 7 + n 3 sin ⁡ 3 π 14 + n 4 sin ⁡ 2 π 7 + n 5 sin ⁡ 5 π 14 + n 6 sin ⁡ 3 π 7 + n 7 sin ⁡ π 2 \small 7\cot\frac{\pi}{28} = n_1\sin\frac{\pi}{14} + n_2\sin\frac{\pi}{7} + n_3\sin\frac{3\pi}{14} + n_4 \sin\frac{2\pi}{7} + n_5 \sin\frac{5\pi}{14} + n_6 \sin\frac{3\pi}{7} + n_7\sin\frac{\pi}{2}

How many distinct nonnegative integer solutions of ( n 1 , n 2 , n 3 , n 4 , n 5 , n 6 , n 7 ) (n_1, n_2, n_3, n_4, n_5, n_6, n_7) satisfy the equation?

Inspiration: (1) , (2) , (3)

You might want to try the three linked problems first, which will (in)definitely help.


The answer is 15.

Michael Huang by Michael Huang , 29, USA

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1 solution

Yuriy Kazakov
Apr 11, 2021

Here there are examples of solutions.

Example 1

Example 2

Example 3

I use Phyton.

Solution give fine trigonometry equation 1 2 − sin ⁡ 5 π 14 − sin ⁡ π 14 + sin ⁡ 3 π 14 = 0 \frac{1}{2} - \sin{\frac{5 \pi}{14}} - \sin{\frac{\pi}{14}} +\sin{\frac{3 \pi}{14}}=0 

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import math

q=math.pi/14
A1=math.sin(q)
B1=math.sin(2*q)
C1=math.sin(3*q)
D1=math.sin(4*q)
E1=math.sin(5*q)
F1=math.sin(6*q)

R = 7/math.tan(q/2)
print(R,A1,B1,C1,D1,E1,F1)
N=0
w=1
for A in range(0,279,w):
    #for B in range(0,math.trunc((R-A*A1)/B1)+1,w):
    for B in range(14,15):
      for C in range(0,math.trunc((R-A*A1-B*B1)/C1)+1,w):
        #for D in range (0, math.trunc((R-A*A1-B*B1-C*C1)/D1)+1,w):
        for D in range(14,15):
          for E in range (0,math.trunc((R-A*A1-B*B1-C*C1-D*D1)/E1)+1,w):
            for F in range(14,15):
            #for F in range(0,math.trunc((R-A*A1-B*B1-C*C1-D*D1-E*E1)/F1)+1,w):
              for G in range(0,math.trunc(R-A*A1-B*B1-C*C1-D*D1-E*E1-F*F1)+1,w):
                  V = A*A1+ B*B1+C*C1+D*D1+E*E1+F*F1+G
                  if abs(R-V)<0.0000001:
                    N=N+1
                    print("N=",N,' A=', A,' B=', B,' C=', C,' D=', D, ' E=', E, ' F=', F, 'G=',G)

print(N)


N= 1    A= 0   B= 14  C= 28  D= 14  E= 0  F= 14  G= 14
N= 2   A= 2   B= 14  C= 26  D= 14   E= 2   F= 14  G= 13
N= 3   A= 4   B= 14  C= 24  D= 14   E= 4  F= 14  G= 12
N= 4   A= 6   B= 14  C= 22  D= 14   E= 6  F= 14  G= 11
N= 5   A= 8   B= 14  C= 20  D= 14   E= 8  F= 14  G= 10
N= 6   A= 10  B= 14  C= 18  D= 14   E= 10  F= 14  G= 9
N= 7    A= 12  B= 14  C= 16  D= 14   E= 12  F= 14  G= 8
N= 8   A= 14  B= 14  C= 14  D= 14   E= 14  F= 14  G= 7
N= 9   A= 16  B= 14  C= 12  D= 14   E= 16  F= 14  G= 6
N= 10  A= 18  B= 14  C= 10  D= 14  E= 18  F= 14  G= 5
N= 11   A= 20  B= 14  C= 8  D= 14   E= 20  F= 14  G= 4
N= 12  A= 22  B= 14  C= 6  D= 14   E= 22   F= 14  G= 3
N= 13  A= 24  B= 14  C= 4  D= 14   E= 24   F= 14  G= 2
N= 14  A= 26  B= 14  C= 2  D= 14   E= 26   F= 14  G= 1
N= 15  A= 28  B= 14  C= 0  D= 14   E= 28   F= 14  G= 0
15

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