Puzzle for November 12, 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.5, replace B + C with A (from eq.2): A + D = A – C + F In the above equation, subtract A from both sides, and add C to both sides: A + D – A + C = A – C + F – A + C which becomes D + C = F which may be written as eq.5a) C + D = F
Hint #2
In eq.4, replace F with C + D (from eq.5a): C – D + E = D + C + D which becomes C – D + E = 2×D + C In the above equation, subtract C from both sides, and add D to both sides: C – D + E – C + D = 2×D + C – C + D which simplifies to E = 3×D
Hint #3
In eq.3, substitute B + C for A (from eq.2), and 3×D for E: B + C + C = B + D + 3×D which becomes B + 2×C = B + 4×D Subtract B from both sides of the above equation: B + 2×C – B = B + 4×D – B which makes 2×C = 4×D Divide both sides by 2: 2×C ÷ 2 = 4×D ÷ 2 which makes C = 2×D
Hint #4
Substitute 2×D for C in eq.2: eq.2a) B + 2×D = A
Hint #5
Substitute 2×D for C in eq.5a: 2×D + D = F which makes 3×D = F
Hint #6
Substitute 3×D for E and F in eq.6: 3×D + 3×D – A – B = A + D which becomes 6×D – A – B = A + D In the above equation, add A to both sides, and subtract D from both sides: 6×D – A – B + A – D = A + D + A – D which becomes eq.6a) 5×D – B = 2×A
Hint #7
Substitute (B + 2×D) for A (from eq.2a) in eq.6a: 5×D – B = 2×(B + 2×D) which becomes 5×D – B = 2×B + 4×D In the above equation, add B to both sides, and subtract 4×D from both sides: 5×D – B + B – 4×D = 2×B + 4×D + B – 4×D which becomes D = 3×B Divide both sides by 3: D ÷ 3 = 3×B ÷ 3 which makes ⅓×D = B
Hint #8
Substitute ⅓×D for B into eq.2a: ⅓×D + 2×D = A which makes 2⅓×D = A
Solution
Substitute 2⅓×D for A, ⅓×D for B, 2×D for C, and 3×D for E and F in eq.1: 2⅓×D + ⅓×D + 2×D + D + 3×D + 3×D = 35 which simplifies to 11⅔×D = 35 Divide both sides of the above equation by 11⅔: 11⅔×D ÷ 11⅔ = 35 ÷ 11⅔ which means D = 3 making A = 2⅓×D = 2⅓ × 3 = 7 B = ⅓×D = ⅓ × 3 = 1 C = 2×D = 2 × 3 = 6 E = F = 3×D = 3 × 3 = 9 and ABCDEF = 716399