Puzzle for November 28, 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.3, replace B with C + E (from eq.2): D + E = C + E + C which becomes D + E = 2×C + E Subtract E from both sides of the equation above: D + E - E = 2×C + E - E which makes D = 2×C
Hint #2
In eq.4, replace D with 2×C: C - A = A - 2×C Add A and 2×C to both sides of the above equation: C - A + A + 2×C = A - 2×C + A + 2×C which becomes 3×C = 2×A Divide both sides by 2: 3×C ÷ 2 = 2×A ÷ 2 which makes 1½×C = A
Hint #3
In eq.5, substitute 1½×C for A: F - C = 1½×C - F Add C and F to both sides of the equation above: F - C + C + F = 1½×C - F + C + F which makes 2×F = 2½×C Divide both sides by 2: 2×F ÷ 2 = 2½×C ÷ 2 which makes F = 1¼×C
Hint #4
Substitute 1½×C for A, and 1¼×C for F in eq.6: 1½×C + 1¼×C = B + C which becomes 2¾×C = B + C Subtract C from each side of the equation above: 2¾×C - C = B + C - C which makes 1¾×C = B
Hint #5
Substitute 1¾×C for B in eq.2: 1¾×C = C + E Subtract C from each side of the above equation: 1¾×C - C = C + E - C which makes ¾×C = E
Solution
Substitute 1½×C for A, 1¾×C for B, 2×C for D, ¾×C for E, and 1¼×C for F in eq.1: 1½×C + 1¾×C + C + 2×C + ¾×C + 1¼×C = 33 which simplifies to 8¼×C = 33 Divide both sides of the above equation by 2: 8¼×C ÷ 8¼ = 33 ÷ 8¼ which means C = 4 making A = 1½×C = 1½ × 4 = 6 B = 1¾×C = 1¾ × 4 = 7 D = 2×C = 2 × 4 = 8 E = ¾×C = ¾ × 4 = 3 F = 1¼×C = 1¼ × 4 = 5 and ABCDEF = 674835