Puzzle for May 27, 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
In eq.4, replace F with B + E (from eq.2): A + B = D + E + B + E which becomes A + B = D + 2×E + B Subtract B from both sides of the above equation: A + B – B = D + 2×E + B – B which becomes eq.4a) A = D + 2×E
Hint #2
In eq.6, replace A with D + 2×E (from eq.4a): D – C + E = D + 2×E + C – E which becomes D – C + E = D + E + C In the above equation, subtract D and E from each side, and add C to each side: D – C + E – D – E + C = D + E + C – D – E + C which simplifies to 0 = 2×C which means 0 = C
Hint #3
In eq.3, substitute D + 2×E for A (from eq.4a), and 0 for C: B + D = D + 2×E – 0 which becomes B + D = D + 2×E Subtract D from each side of the above equation: B + D – D = D + 2×E – D which makes eq.3a) B = 2×E
Hint #4
Substitute 0 for C in eq.5: 0 – D + F = B + D which becomes –D + F = B + D Add D to both sides of the above equation: –D + F + D = B + D + D which becomes eq.5a) F = B + 2×D
Hint #5
Substitute B + 2×D for F (from eq.5a) in eq.4: A + B = D + E + B + 2×D which becomes A + B = E + B + 3×D Subtract B from each side of the equation above: A + B – B = E + B + 3×D – B which becomes eq.4b) A = E + 3×D
Hint #6
Substitute 3×D + E for A (from eq.4b) into eq.4a: E + 3×D = D + 2×E Subtract E and D from each side of the above equation: E + 3×D – E – D = D + 2×E – E – D which makes 2×D = E
Hint #7
Substitute (2×D) for E in eq.3a: B = 2×(2×D) which makes B = 4×D
Hint #8
Substitute 2×D for E in eq.4b: A = 2×D + 3×D which makes A = 5×D
Hint #9
Substitute 4×D for B, and 2×D for E in eq.2: F = 4×D + 2×D which makes F = 6×D
Solution
Substitute 5×D for A, 4×D for B, 0 for C, 2×D for E and 6×D for F in eq.1: 5×D + 4×D + 0 + D + 2×D + 6×D = 18 which simplifies to 18×D = 18 Divide both sides of the above equation by 18: 18×D ÷ 18 = 18 ÷ 18 which means D = 1 making A = 5×D = 5 × 1 = 5 B = 4×D = 4 × 1 = 4 E = 2×D = 2 × 1 = 2 F = 6×D = 6 × 1 = 6 and ABCDEF = 540126