Puzzle for December 23, 2021 ( )
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
Add F to both sides of eq.6: A + B + F + F = C + D – F + F which becomes A + B + 2×F = C + D In eq.5, replace C + D with A + B + 2×F: A + B + 2×F + F = B – C + E which becomes A + B + 3×F = B – C + E In the above equation, subtract B from both sides, and add C to both sides: A + B + 3×F – B + C = B – C + E – B + C which becomes eq.5a) A + 3×F + C = E
Hint #2
In eq.3, replace E with A + 3×F + C (from eq.5a): A + 3×F + C + F = A + C which becomes A + 4×F + C = A + C Subtract A and C from both sides of the above equation: A + 4×F + C – A – C = A + C – A – C which simplifies to 4×F = 0 which means F = 0
Hint #3
In eq.4, substitute 0 for F: B + D = C – D + 0 which becomes B + D = C – D Add D to both sides of the equation above: B + D + D = C – D + D which becomes eq.4a) B + 2×D = C
Hint #4
Substitute 0 for F, and B + 2×D for C (from eq.4a) in eq.6: A + B + 0 = B + 2×D + D – 0 which becomes A + B = B + 3×D Subtract B from each side of the equation above: A + B – B = B + 3×D – B which makes A = 3×D
Hint #5
Substitute 0 for F, and 3×D for A in eq.3: E + 0 = 3×D + C which becomes eq.3a) E = 3×D + C
Hint #6
Substitute 3×D + C for E (from eq.3a) into eq.2: D + 3×D + C = B + C which becomes 4×D + C = B + C Subtract C from each side of the equation above: 4×D + C – C = B + C – C which makes 4×D = B
Hint #7
Substitute 4×D for B in eq.4a: 4×D + 2×D = C which makes 6×D = C
Hint #8
Substitute 6×D for C in eq.3a: E = 3×D + 6×D which makes E = 9×D
Solution
Substitute 3×D for A, 4×D for B, 6×D for C, 9×D for E, and 0 for F in eq.1: 3×D + 4×D + 6×D + D + 9×D + 0 = 23 which simplifies to 23×D = 23 Divide both sides of the above equation by 23: 23×D ÷ 23 = 23 ÷ 23 which means D = 1 making A = 3×D = 3 × 1 = 3 B = 4×D = 4 × 1 = 4 C = 6×D = 6 × 1 = 6 E = 9×D = 9 × 1 = 9 and ABCDEF = 346190