Puzzle for December 14, 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.4, replace F with C + D (from eq.3): C + C + D = A + D which becomes 2×C + D = A + D Subtract D from each side of the above equation: 2×C + D – D = A + D – D which makes 2×C = A
Hint #2
In eq.5, replace C + F with A + D (from eq.4): D + E = A + A + D which becomes D + E = 2×A + D Subtract D from both sides of the above equation: D + E – D = 2×A + D – D which makes eq.5a) E = 2×A
Hint #3
In eq.5a, substitute (2×C) for A: E = 2×(2×C) which makes E = 4×C
Hint #4
Substitute F – A for B (from eq.2) in eq.6: E + F – D = F – A + C + D In the above equation, subtract F from both sides, and add D to both sides: E + F – D – F + D = F – A + C + D – F + D which becomes eq.6a) E = –A + C + 2×D
Hint #5
Substitute 4×C for E, and 2×C for A in eq.6a: 4×C = –2×C + C + 2×D which becomes 4×C = –C + 2×D Add C to both sides of the equation above: 4×C + C = –C + 2×D + C which makes 5×C = 2×D Divide both sides by 2: 5×C ÷ 2 = 2×D ÷ 2 which makes 2½×C = D
Hint #6
Substitute 2½×C for D, 4×C for E, and 2×C for A in eq.5: 2½×C + 4×C = 2×C + C + F which becomes 6½×C = 3×C + F Subtract 3×C from each side of the equation above: 6½×C – 3×C = 3×C + F – 3×C which makes 3½×C = F
Hint #7
Substitute 3½×C for F, and 2×C for A in eq.2: B = 3½×C – 2×C which makes B = 1½×C
Solution
Substitute 2×C for A, 1½×C for B, 2½×C for D, 4×C for E, and 3½×C for F in eq.1: 2×C + 1½×C + C + 2½×C + 4×C + 3½×C = 29 which simplifies to 14½×C = 29 Divide both sides of the above equation by 14½: 14½×C ÷ 14½ = 29 ÷ 14½ which means C = 2 making A = 2×C = 2 × 2 = 4 B = 1½×C = 1½ × 2 = 3 D = 2½×C = 2½ × 2 = 5 E = 4×C = 4 × 2 = 8 F = 3½×C = 3½ × 2 = 7 and ABCDEF = 432587