Puzzle for September 15, 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.
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Hint #1
Add E to both sides of eq.2: B - E + E = E + F + E which becomes eq.2a) B = 2×E + F Add C and E to both sides of eq.4: A - E + C + E = E - C + C + E which becomes eq.4a) A + C = 2×E
Hint #2
In eq.2a, replace 2×E with A + C (from eq.4a): eq.2b) B = A + C + F
Hint #3
In eq.3, replace B with A + C + F (from eq.2b): C + F = A + C + F - D In the above equation, subtract C and F from both sides, and add D to both sides: C + F - C - F + D = A + C + F - D - C - F + D which simplifies to D = A
Hint #4
In eq.5, replace B with A + C + F (from eq.2b): A + C + F + C = A + E + F which becomes A + 2×C + F = A + E + F Subtract A and F from each side of the equation above: A + 2×C + F - A - F = A + E + F - A - F which simplifies to 2×C = E
Hint #5
In eq.4a, substitute (2×C) for E: A + C = 2×(2×C) which becomes A + C = 4×C Subtract C from each side of the above equation: A + C - C = 4×C - C which makes A = 3×C and also makes D = A = 3×C
Hint #6
Substitute 3×C for A in eq.2b: B = 3×C + C + F which becomes eq.2c) B = 4×C + F
Hint #7
eq.6 may be written as: C + D = (B + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × (C + D) = 3 × (B + E + F) ÷ 3 which becomes eq.6a) 3×C + 3×D = B + E + F
Hint #8
Substitute (3×C) for D, 4×C + F for B (from eq.2c), and 2×C for E in eq.6a: 3×C + 3×(3×C) = 4×C + F + 2×C + F which becomes 3×C + 9×C = 6×C + 2×F which becomes 12×C = 6×C + 2×F Subtract 6×C from each side of the equation above: 12×C - 6×C = 6×C + 2×F - 6×C which becomes 6×C = 2×F Divide both sides by 2: 6×C ÷ 2 = 2×F ÷ 2 which makes 3×C = F
Hint #9
Substitute 3×C for F in eq.2c: B = 4×C + 3×C which makes B = 7×C
Solution
Substitute 3×C for A and D and F, 7×C for B, and 2×C for E in eq.1: 3×C + 7×C + C + 3×C + 2×C + 3×C = 19 which simplifies to 19×C = 19 Divide both sides of the above equation by 19: 19×C ÷ 19 = 19 ÷ 19 which means C = 1 making A = D = F = 3×C = 3 × 1 = 3 B = 7×C = 7 × 1 = 7 E = 2×C = 2 × 1 = 2 and ABCDEF = 371323