Puzzle for May 6, 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.6: C + D + D = A – D + E + D which becomes eq.6a) C + 2×D = A + E Add B and D to both sides of eq.2: D – B + B + D = C – D + B + D which becomes eq.2a) 2×D = C + B
Hint #2
In eq.6a, replace 2×D with C + B (from eq.2a): C + C + B = A + E which becomes eq.6b) 2×C + B = A + E
Hint #3
In eq.5, replace A + E with 2×C + B (from eq.6b): 2×C + B = B + C + F Subtract C and B from both sides of the equation above: 2×C + B – C – B = B + C + F – C – B which simplifies to C = F
Hint #4
In eq.4, substitute C for F: C – D = E – C Add C to both sides of the equation above: C – D + C = E – C + C which becomes eq.4a) 2×C – D = E
Hint #5
Subtract C from each side of eq.2a: 2×D – C = C + B – C which becomes eq.2b) 2×D – C = B Substitute 2×D – C for B (from eq.2b), and 2×C – D for E (from eq.4a) in eq.6b: 2×C + 2×D – C = A + 2×C – D which becomes C + 2×D = A + 2×C – D In the above equation, subtract 2×C from both sides, and add D to both sides: C + 2×D – 2×C + D = A + 2×C – D – 2×C + D which becomes eq.6c) 3×D – C = A
Hint #6
Substitute (2×C – D) for E (from eq.4a), C for F, and 3×D – C for A (from eq.6c) in eq.3: (2×C – D) – C = 3×D – C – (2×C – D) which becomes C – D = 3×D – C – 2×C + D which becomes C – D = 4×D – 3×C Add D and 3×C to both sides of the above equation: C – D + D + 3×C = 4×D – 3×C + D + 3×C which makes 4×C = 5×D Divide both sides by 4: 4×C ÷ 4 = 5×D ÷ 4 which makes C = 1¼×D and also makes F = C = 1¼×D
Hint #7
Substitute 1¼×D for C in eq.6c: 3×D – 1¼×D = A which makes 1¾×D = A
Hint #8
Substitute 1¼×D for C in eq.2b: 2×D – 1¼×D = B which makes ¾×D = B
Hint #9
Substitute (1¼×D) for C in eq.4a: 2×(1¼×D) – D = E which becomes 2½×D – D = E which makes 1½×D = E
Solution
Substitute 1¾×D for A, ¾×D for B, 1¼×D for C and F, and 1½×D for E in eq.1: 1¾×D + ¾×D + 1¼×D + D + 1½×D + 1¼×D = 30 which simplifies to 7½×D = 30 Divide both sides of the above equation by 7½: 7½×D ÷ 7½ = 30 ÷ 7½ which means D = 4 making A = 1¾×D = 1¾ × 4 = 7 B = ¾×D = ¾ × 4 = 3 C = F = 1¼×D = 1¼ × 4 = 5 E = 1½×D = 1½ × 4 = 6 and ABCDEF = 735465