Puzzle for September 15, 2020 ( )
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.
* "D ^ C" means "D raised to the power of C".
Scratchpad
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Hint #1
In eq.3, add F to both sides, and subtract B from both sides: C – F + F – B = B – A + F – B which becomes C – B = –A + F which may be written as C – B = F – A In eq.2, replace C – B with F – A: F – E = F – A Subtract F from both sides of the equation above: F – E – F = F – A – F which makes –E = –A which means E = A
Hint #2
In eq.6, replace A with E: D ^ C = E – E which becomes D ^ C = 0 which makes D = 0 and also makes C > 0
Hint #3
In eq.4, substitute 0 for D: B + E = 0 + F which becomes eq.4a) B + E = F
Hint #4
Substitute B + E for F (from eq.4a) in eq.2: B + E – E = C – B which becomes B = C – B Add B to both sides of the equation above: B + B = C – B + B which makes 2×B = C
Hint #5
Substitute E for A, 0 for D, and 2×B for C in eq.5: E + 0 + E = B + 2×B which becomes 2×E = 3×B Divide both sides of the above equation by 2: 2×E ÷ 2 = 3×B ÷ 2 which makes E = 1½×B and also makes A = E = 1½×B
Hint #6
Substitute 1½×B for E, and 2×B for C in eq.2: F – 1½×B = 2×B – B which becomes F – 1½×B = B Add 1½×B to both sides of the equation above: F – 1½×B + 1½×B = B + 1½×B which makes F = 2½×B
Solution
Substitute 1½×B for A and E, 2×B for C, 0 for D, and 2½×B for F in eq.1: 1½×B + B + 2×B + 0 + 1½×B + 2½×B = 17 which simplifies to 8½×B = 17 Divide both sides of the above equation by 8½: 8½×B ÷ 8½ = 17 ÷ 8½ which means B = 2 making A = E = 1½×B = 1½ × 2 = 3 C = 2×B = 2 × 2 = 4 F = 2½×B = 2½ × 2 = 5 and ABCDEF = 324035