Puzzle for January 24, 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
In eq.4, replace D with E + F (from eq.3): C + E = E + F – C In the equation above, subtract E from both sides, and add C to both sides: C + E – E + C = E + F – C – E + C which makes 2×C = F
Hint #2
In eq.6, replace C + D with A (from eq.2): A + B = A + F Subtract A from each side of the above equation: A + B – A = A + F – A which makes B = F and also makes B = F = 2×C
Hint #3
In eq.5, substitute E + F for D (from eq.3), and 2×C for B: E + F + E = 2×C + C + F which becomes 2×E + F = 3×C + F Subtract F from each side of the above equation: 2×E + F – F = 3×C + F – F which makes 3×C = 2×E Divide both sides by 2: 3×C ÷ 2 = 2×E ÷ 2 which makes 1½×C = E
Hint #4
Substitute 1½×C for E, and 2×C for F in eq.3: 1½×C + 2×C = D which makes 3½×C = D
Hint #5
Substitute 3½×C for D in eq.2: C + 3½×C = A which makes 4½×C = A
Solution
Substitute 4½×C for A, 2×C for B and F, 3½×C for D, and 1½×C for E in eq.1: 4½×C + 2×C + C + 3½×C + 1½×C + 2×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 = 4½×C = 4½ × 2 = 9 B = F = 2×C = 2 × 2 = 4 D = 3½×C = 3½ × 2 = 7 E = 1½×C = 1½ × 2 = 3 and ABCDEF = 942734