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 Some summed sin and cos (Posted on 2009-09-28)
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 No Rating

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 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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