Puzzle for August 25, 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
Add D to both sides of eq.3: E – D + D = C + D + D which becomes eq.3a) E = C + 2×D In eq.6, add B to both sides, and subtract D from both sides: D + E – B + B – D = A + B – D + B – D which becomes eq.6a) E = A + 2×B – 2×D
Hint #2
In eq.6a, replace E with C + 2×D (from eq.3a): C + 2×D = A + 2×B – 2×D Subtract 2×D from both sides of the above equation: C + 2×D – 2×D = A + 2×B – 2×D – 2×D which becomes eq.6b) C = A + 2×B – 4×D
Hint #3
Add E to both sides of eq.5: E – C + E = A – B – E + E which becomes 2×E – C = A – B In the above equation, substitute (A + 2×B – 2×D) for E (from eq.6a): 2×(A + 2×B – 2×D) – C = A – B which becomes 2×A + 4×B – 4×D – C = A – B Add C and B to both sides, and subtract A from both sides: 2×A + 4×B – 4×D – C + C + B – A = A – B + C + B – A which becomes eq.5a) A + 5×B – 4×D = C
Hint #4
In eq.6b, replace C with A + 5×B – 4×D (from eq.5a): A + 5×B – 4×D = A + 2×B – 4×D In the above equation, subtract A from both sides, and add 4×D to both sides: A + 5×B – 4×D – A + 4×D = A + 2×B – 4×D – A + 4×D which makes 5×B = 2×B Subtract 2×B from each side: 5×B – 2×B = 2×B – 2×B which makes 3×B = 0 which means B = 0
Hint #5
In eq.2, substitute 0 for B, and C + 2×D for E (from eq.3a): 0 + C + 2×D = C + F which becomes C + 2×D = C + F Subtract C from each side of the above equation: C + 2×D – C = C + F – C which makes 2×D = F
Hint #6
Substitute 2×D for F in eq.4: C – 2×D = D + 2×D which becomes C – 2×D = 3×D Add 2×D to both sides of the equation above: C – 2×D + 2×D = 3×D + 2×D which makes C = 5×D
Hint #7
Substitute 5×D for C in eq.3a: E = 5×D + 2×D which makes E = 7×D
Hint #8
Substitute 5×D for C, and 0 for B in eq.6b: 5×D = A + 2×0 – 4×D which becomes 5×D = A + 0 – 4×D which becomes 5×D = A – 4×D Add 4×D to both sides of the above equation: 5×D + 4×D = A – 4×D + 4×D which makes 9×D = A
Solution
Substitute 9×D for A, 0 for B, 5×D for C, 7×D for E, and 2×D for F in eq.1: 9×D + 0 + 5×D + D + 7×D + 2×D = 24 which simplifies to 24×D = 24 Divide both sides of the above equation by 24: 24×D ÷ 24 = 24 ÷ 24 which means D = 1 making A = 9×D = 9 × 1 = 9 C = 5×D = 5 × 1 = 5 E = 7×D = 7 × 1 = 7 F = 2×D = 2 × 1 = 2 and ABCDEF = 905172