Puzzle for July 29, 2019 ( )
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 E with B + C (from eq.5): B – C = B + C – D Subtract B from each side of the above equation: B – C – B = B + C – D – B which becomes –C = C – D Add C + D to each side: –C + C + D = C – D + C + D which means D = 2×C
Hint #2
Add A to both sides of eq.6: D – A + A = A + C + A which becomes D = 2×A + C In the above equation, replace D with 2×C: 2×C = 2×A + C Subtract C from both sides: 2×C – C = 2×A + C – C which makes C = 2×A Divide both sides by 2: C ÷ 2 = 2×A ÷ 2 which means ½×C = A
Hint #3
Substitute 2×C for D, and ½×C for A in eq.4: C + 2×C = B – ½×C Add ½×C to both sides of the above equation: C + 2×C + ½×C = B – ½×C + ½×C which makes 3½×C = B
Hint #4
Substitute 3½×C for B in eq.5: E = 3½×C + C which makes E = 4½×C
Hint #5
Substitute ½×C for A, and 3½×C for B in eq.2: F – ½×C = ½×C + 3½×C which becomes F – ½×C = 4×C Add ½×C to each side: F – ½×C + ½×C = 4×C + ½×C which makes F = 4½×C
Solution
Substitute ½×C for A, 3½×C for B, 2×C for D, and 4½×C for E and F in eq.1: ½×C + 3½×C + C + 2×C + 4½×C + 4½×C = 32 which simplifies to 16×C = 32 Divide both sides of the above equation by 16: 16×C ÷ 16 = 32 ÷ 16 which means C = 2 making A = ½×C = ½ × 2 = 1 B = 3½×C = 3½ × 2 = 7 D = 2×C = 2 × 2 = 4 E = F = 4½×C = 4½ × 2 = 9 and ABCDEF = 172499