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Some summed sin and cos (Posted on 2009-09-28) Difficulty: 2 of 5
Substitute each of the capital letters in bold by a different digit from 0 to 9 to satisfy this system of equations. All the angles are in degrees and none of the numbers can contain any leading zero.
  • cos NE + cos DG = cos AE, and:
  • sin SH + sin AH = sin NH, and:
  • cos RR + cos TS = cos R, and:
  • sin ST + sin ET = sin OT.

See The Solution Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2

The boldface numbers in the same order shown in the puzzle:

81  39  21
40  20  80
66  54  6
45  15  75

DEFDBL A-Z
dr = ATN(1) / 45
CLS

FOR n = 1 TO 9
 IF used(n) = 0 THEN
  used(n) = 1
FOR d = 1 TO 9
 IF used(d) = 0 THEN
  used(d) = 1
FOR a = 1 TO 9
 IF used(a) = 0 THEN
  used(a) = 1
FOR s = 1 TO 9
 IF used(s) = 0 THEN
  used(s) = 1
FOR r = 1 TO 9
 IF used(r) = 0 THEN
  used(r) = 1
  rr = r * 10 + r
FOR e = 1 TO 9
 IF used(e) = 0 THEN
  used(e) = 1
  ne = n * 10 + e
  ae = a * 10 + e
FOR t = 1 TO 9
 IF used(t) = 0 THEN
  used(t) = 1
  ts = t * 10 + s
  st = s * 10 + t
  et = e * 10 + t
FOR o = 1 TO 9
 IF used(o) = 0 THEN
  used(o) = 1
  ot = o * 10 + t
FOR g = 0 TO 9
 IF used(g) = 0 THEN
  used(g) = 1
  dg = d * 10 + g
FOR h = 0 TO 9
 IF used(h) = 0 THEN
  used(h) = 1
 
  nh = n * 10 + h
  sh = s * 10 + h
  ah = a * 10 + h

  lhs = COS(ne * dr) + COS(dg * dr): rhs = COS(ae * dr)
  IF ABS(lhs - rhs) < 1E-08 THEN

  lhs = SIN(sh * dr) + SIN(ah * dr): rhs = SIN(nh * dr)
  IF ABS(lhs - rhs) < 1E-08 THEN

  lhs = COS(rr * dr) + COS(ts * dr): rhs = COS(r * dr)
  IF ABS(lhs - rhs) < 1E-08 THEN

  lhs = SIN(st * dr) + SIN(et * dr): rhs = SIN(ot * dr)
  IF ABS(lhs - rhs) < 1E-08 THEN

  PRINT ne; dg; ae
  PRINT sh; ah; nh
  PRINT rr; ts; r
  PRINT st; et; ot
  PRINT


  END IF

  END IF

  END IF

  END IF

  used(h) = 0
 END IF
NEXT
  used(g) = 0
 END IF
NEXT
  used(o) = 0
 END IF
NEXT
  used(t) = 0
 END IF
NEXT
  used(e) = 0
 END IF
NEXT
  used(r) = 0
 END IF
NEXT
  used(s) = 0
 END IF
NEXT
  used(a) = 0
 END IF
NEXT
  used(d) = 0
 END IF
NEXT
  used(n) = 0
 END IF
NEXT

 


  Posted by Charlie on 2009-09-28 18:57:28
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