Puzzle for August 19, 2020 ( )
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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.
* "D ^ E" means "D raised to the power of E".
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Hint #1
Add F to both sides of eq.4: C + E + F = A – F + F which becomes eq.4a) C + E + F = A Add B to both sides of eq.3: D – B + B = C + B which becomes eq.3a) D = C + B
Hint #2
In eq.2, replace A with C + E + F (from eq.4a), and D with C + B (from eq.3a): C + E + F + B + C + E = C + B + F – C – E which becomes 2×C + 2×E + F + B = B + F – E In the above equation, subtract B and F from each side, and add E to each side: 2×C + 2×E + F + B – B – F + E = B + F – E – B – F + E which simplifies to 2×C + 3×E = 0 Since C and E are non-negative, the above equation makes: C = 0 and E = 0
Hint #3
In eq.4a, replace C and E with 0: 0 + 0 + F = A which makes F = A
Hint #4
In eq.3a, replace C with 0: D = 0 + B which makes D = B
Hint #5
Substitute 0 for E in eq.5: F = D ^ 0 which makes 1 = D ^ 0 (implies D ≠ 0) making F = 1 and also makes A = F = 1
Solution
Substitute 1 for A and F, 0 for C and E, and B for D in eq.1: 1 + B + 0 + B + 0 + 1 = 10 which becomes 2 + 2×B = 10 Subtract 2 from both sides of the equation above: 2 + 2×B – 2 = 10 – 2 which makes 2×B = 8 Divide both sides by 2: 2×B ÷ 2 = 8 ÷ 2 which means B = 4 making D = B = 4 and ABCDEF = 140401