Puzzle for January 8, 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.
* "F ^ A" means "F raised to the power of A". "A ^ C" means "A raised to the power of C".
Scratchpad
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Hint #1
Add C + E to both sides of eq.3: B - E + C + E = F - C + C + E which becomes B + C = F + E which may be written as B + C = E + F In eq.2, replace E + F with B + C: A + B = D + B + C Subtract B from each side of the above equation: A + B - B = D + B + C - B which reduces to eq.3a) A = D + C
Hint #2
In eq.4, replace A with D + C (from eq.3a): D + C + C = D - C Add (C - D) to each side: D + C + C + (C - D) = D - C + (C - D) which simplifies to 3×C = 0 which means C = 0
Hint #3
In eq.4, replace C with 0: A + 0 = D - 0 which becomes A = D
Hint #4
In eq.6, substitute 0 for C: F ^ A = A ^ 0 which makes F ^ A = 1 (this presumes A ≠ 0) To make the above equation true, then: A ≥ 0 and F = 1 or: A = 0 and F > 0 Since A ≠ 0, then A > 0 and F = 1
Hint #5
Substitute 0 for C, and 1 for F in eq.3: B - E = 1 - 0 which becomes B - E = 1 Add E to both sides of the equation above: B - E + E = 1 + E which becomes eq.3b) B = 1 + E
Hint #6
Substitute 1 + E for B (from eq.3b), A for D, 0 for C, and 1 for F into eq.5: 1 + E - A = A - 0 + 1 Add (A - 1) to each side: 1 + E - A + (A - 1) = A - 0 + 1 + (A - 1) which simplifies to E = 2×A
Hint #7
Substitute 2×A for E in eq.3b: eq.3c) B = 1 + 2×A
Solution
Substitute 1 + 2×A for B (from eq.3c), 0 for C, A for D, 2×A for E, and 1 for F in eq.1: A + 1 + 2×A + 0 + A + 2×A + 1 = 26 which simplifies to 6×A + 2 = 26 Subtract 2 from each side: 6×A + 2 - 2 = 26 - 2 which becomes 6×A = 24 Divide both sides by 6: 6×A ÷ 6 = 24 ÷ 6 which makes A = 4 making B = 1 + 2×A = 1 + 2×4 = 1 + 8 = 9 D = A = 4 E = 2×A = 2×4 = 8 and ABCDEF = 490481