Puzzle for October 11, 2022 ( )
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.
Scratchpad
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Hint #1
Subtract F from both sides of eq.5: E - F - F = D + F - F which becomes eq.5a) E - 2×F = D
Hint #2
In eq.6, substitute (E - 2×F) for D (from eq.5a): F - (E - 2×F) = (E - 2×F) - E which becomes F - E + 2×F = -2×F which becomes 3×F - E = -2×F Add E and 2×F to both sides of the above equation: 3×F - E + E + 2×F = -2×F + E + 2×F which makes 5×F = E
Hint #3
In eq.5a, replace E with 5×F: 5×F - 2×F = D which makes 3×F = D
Hint #4
In eq.2, replace D with 3×F: 3×F = A + F Subtract F from each side of the equation above: 3×F - F = A + F - F which makes 2×F = A
Hint #5
In eq.4, substitute 5×F for E: C + F = B + 5×F Subtract F from each side of the equation above: C + F - F = B + 5×F - F which becomes eq.4a) C = B + 4×F
Hint #6
Substitute B + 4×F for C (from eq.4a), 3×F for D, and 5×F for E in eq.3: B + B + 4×F = 3×F + 5×F which becomes 2×B + 4×F = 8×F Subtract 4×F from each side of the above equation: 2×B + 4×F - 4×F = 8×F - 4×F which becomes 2×B = 4×F Divide both sides by 2: 2×B ÷ 2 = 4×F ÷ 2 which makes B = 2×F
Hint #7
Substitute 2×F for B in eq.4a: C = 2×F + 4×F which makes C = 6×F
Solution
Substitute 2×F for A and B, 6×F for C, 3×F for D, and 5×F for E into eq.1: 2×F + 2×F + 6×F + 3×F + 5×F + F = 19 which simplifies to 19×F = 19 Divide both sides of the above equation by 19: 19×F ÷ 19 = 19 ÷ 19 which means F = 1 making A = B = 2×F = 2 × 1 = 2 C = 6×F = 6 × 1 = 6 D = 3×F = 3 × 1 = 3 E = 5×F = 5 × 1 = 5 and ABCDEF = 226351