Puzzle for May 29, 2023 ( )
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 F with B + C (from eq.2): B + C - B = B - C which becomes C = B - C Add C to both sides of the above equation: C + C = B - C + C which makes 2×C = B
Hint #2
In eq.2, replace B with 2×C: F = 2×C + C which makes F = 3×C
Hint #3
In eq.6, replace B with 2×C: 2×C + E - C = C + D which becomes C + E = C + D Subtract C from each side of the equation above: C + E - C = C + D - C which makes E = D
Hint #4
In eq.4, replace B with 2×C, and E with D: 2×C - D = C + D In the above equation, add D to both sides, and subtract C from both sides: 2×C - D + D - C = C + D + D - C which makes C = 2×D Divide both sides by 2: C ÷ 2 = 2×D ÷ 2 which makes ½×C = D and also makes ½×C = D = E
Hint #5
In eq.3, substitute ½×C for E: C = A + ½×C Subtract ½×C from each side of the above equation: C - ½×C = A + ½×C - ½×C which makes ½×C = A
Solution
Substitute ½×C for A and D and E, 2×C for B, and 3×C for F in eq.1: ½×C + 2×C + C + ½×C + ½×C + 3×C = 15 which simplifies to 7½×C = 15 Divide both sides of the above equation by 7½: 7½×C ÷ 7½ = 15 ÷ 7½ which means C = 2 making A = D = E = ½×C = ½ × 2 = 1 B = 2×C = 2 × 2 = 4 F = 3×C = 3 × 2 = 6 and ABCDEF = 142116