Puzzle for September 22, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* "C ^ E" means "C raised to the power of E".
** DE is a 2-digit number (not D×E). DE is an angle expressed in degrees.
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Hint #1
In eq.4, the only non-negative two-digit integers that yield an integer value for secant are: secant (00) = 1 making D = 0 and E = 0 or secant (60) = 2 making D = 6 and E = 0 In either of these two cases: E = 0
Hint #2
In eq.3, replace E with 0: D – A = C ^ 0 which becomes D – A = 1 (assumes C ≠ 0) In the above equation, add A to both sides, and subtract 1 from each side: D – A + A – 1 = 1 + A – 1 which makes eq.3a) D – 1 = A
Hint #3
To make eq.3a true, check the two possible values for D (0 or 6): If D = 0, then A = 0 – 1 = –1 If D = 6, then A = 6 – 1 = 5 Since A must be a non-negative integer, this means: A = 5 making D = 6
Hint #4
Substitute 6 for D, and 0 for E in eq.4: secant (60) = B which makes secant (60) = 2 which means B = 2
Hint #5
Substitute 2 for B, and 6 for D in eq.2: 2 + F = 6 Subtract 2 from both sides of the above equation: 2 + F – 2 = 6 – 2 which makes F = 4
Solution
Substitute 2 for B, 0 for E, and 4 for F in eq.1: 2 + C = 0 + 4 Subtract 2 from both sides of the equation above: 2 + C – 2 = 0 + 4 – 2 which makes C = 2 and ABCDEF = 522604