Puzzle for February 4, 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
In eq.4, replace A with B + C (from eq.2): C + E = B + C + D Subtract C from each side of the equation above: C + E – C = B + C + D – C which becomes eq.4a) E = B + D
Hint #2
In eq.3, replace E with B + D (from eq.4a): B + F = D + B + D which becomes B + F = 2×D + B Subtract B from each side of the above equation: B + F – B = 2×D + B – B which makes F = 2×D
Hint #3
In eq.6, replace A with B + C (from eq.2): B + E + F = B + C – E – F In the equation above, subtract B from both sides, and add E and F to both sides: B + E + F – B + E + F = B + C – E – F – B + E + F which simplifies to eq.6a) 2×E + 2×F = C
Hint #4
In eq.5, substitute 2×E + 2×F for C (from eq.6a): D + E = 2×E + 2×F – B – E – F which becomes D + E = E + F – B In the above equation, subtract E from both sides, and add B to both sides: D + E – E + B = E + F – B – E + B which becomes D + B = F which may be written as eq.5a) B + D = F
Hint #5
Substitute E for B + D (from eq.4a), and 2×D for F in eq.5a: E = 2×D
Hint #6
Substitute 2×D for F and E in eq.3: B + 2×D = D + 2×D which becomes B + 2×D = 3×D Subtract 2×D from each side of the equation above: B + 2×D – 2×D = 3×D – 2×D which makes B = D
Hint #7
Substitute (2×D) for E and F in eq.6a: 2×(2×D) + 2×(2×D) = C which becomes 4×D + 4×D = C which makes 8×D = C
Hint #8
Substitute D for B, and 8×D for C in eq.2: A = D + 8×D which makes A = 9×D
Solution
Substitute 9×D for A, D for B, 8×D for C, and 2×D for E and F in eq.1: 9×D + D + 8×D + D + 2×D + 2×D = 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 = 9×D = 9 × 1 = 9 B = D = 1 C = 8×D = 8 × 1 = 8 E = F = 2×D = 2 × 1 = 2 and ABCDEF = 918122