Puzzle for July 8, 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.3, replace A + C with E + F – A (from eq.5): B – C = E + F – A – F which becomes B – C = E – A Add C and A to both sides of the above equation: B – C + C + A = E – A + C + A which becomes B + A = E + C which may be written as eq.5a) A + B = C + E
Hint #2
In eq.4, replace A + B with C + E (from eq.5a): D + E = C + E – D In the equation above, subtract E from both sides, and add D to both sides: D + E – E + D = C + E – D – E + D which simplifies to 2×D = C
Hint #3
In eq.3, substitute 2×D for C: B – 2×D = A + 2×D – F Add 2×D and F to both sides of the equation above: B – 2×D + 2×D + F = A + 2×D – F + 2×D + F which becomes eq.3a) B + F = A + 4×D
Hint #4
Add B to both sides of eq.6: B + F + B = A – B + D + E + B which becomes 2×B + F = A + D + E which may be written as eq.6a) B + B + F = A + D + E
Hint #5
Substitute A + 4×D for B + F (from eq.3a) in eq.6a: B + A + 4×D = A + D + E Subtract A and D from each side of the equation above: B + A + 4×D – A – D = A + D + E – A – D which simplifies to eq.6b) B + 3×D = E
Hint #6
In eq.2, substitute 2×D for C, and B + 3×D for E (from eq.6b): 2×D + D = B + B + 3×D which becomes 3×D = 2×B + 3×D Subtract 3×D from each side of the above equation: 3×D – 3×D = 2×B + 3×D – 3×D which makes 0 = 2×B which means 0 = B
Hint #7
Substitute 0 for B in eq.6b: 0 + 3×D = E which makes 3×D = E
Hint #8
Substitute 0 for B, 2×D for C, and 3×D for E into eq.5a: A + 0 = 2×D + 3×D which makes A = 5×D
Hint #9
Substitute 0 for B, and 5×D for A in eq.3a: 0 + F = 5×D + 4×D which makes F = 9×D
Solution
Substitute 5×D for A, 0 for B, 2×D for C, 3×D for E, and 9×D for F in eq.1: 5×D + 0 + 2×D + D + 3×D + 9×D = 20 which simplifies to 20×D = 20 Divide both sides of the above equation by 20: 20×D ÷ 20 = 20 ÷ 20 which means D = 1 making A = 5×D = 5 × 1 = 5 C = 2×D = 2 × 1 = 2 E = 3×D = 3 × 1 = 3 F = 9×D = 9 × 1 = 9 and ABCDEF = 502139