Puzzle for February 13, 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
Add F and C to both sides of eq.3: C - F + F + C = D - C + F + C which becomes 2×C = D + F In the equation above, replace D + F with B (from eq.2): 2×C = B
Hint #2
In eq.4, substitute 2×C for B: E - C = 2×C - E Add C and E to both sides of the above equation: E - C + C + E = 2×C - E + C + E which makes 2×E = 3×C Divide both sides by 2: 2×E ÷ 2 = 3×C ÷ 2 which makes E = 1½×C
Hint #3
In eq.5, substitute 2×C for B, and 1½×C for E: 2×C + C = A + 1½×C which becomes 3×C = A + 1½×C Subtract 1½×C from each side of the equation above: 3×C - 1½×C = A + 1½×C - 1½×C which makes 1½×C = A
Hint #4
Substitute 1½×C for A, and 2×C for B in eq.6: 1½×C + D = 2×C + C - D which becomes 1½×C + D = 3×C - D In the above equation, subtract 1½×C from both sides, and add D to both sides: 1½×C + D - 1½×C + D = 3×C - D - 1½×C + D which becomes 2×D = 1½×C Divide both sides by 2: 2×D ÷ 2 = 1½×C ÷ 2 which makes D = ¾×C
Hint #5
Substitute 2×C for B, and ¾×C for D in eq.2: 2×C = ¾×C + F Subtract ¾×C from each side of the above equation: 2×C - ¾×C = ¾×C + F - ¾×C which makes 1¼×C = F
Solution
Substitute 1½×C for A and E, 2×C for B, ¾×C for D, and 1¼×C for F in eq.1: 1½×C + 2×C + C + ¾×C + 1½×C + 1¼×C = 32 which simplifies to 8×C = 32 Divide both sides of the above equation by 8: 8×C ÷ 8 = 32 ÷ 8 which means C = 4 making A = E = 1½×C = 1½ × 4 = 6 B = 2×C = 2 × 4 = 8 D = ¾×C = ¾ × 4 = 3 F = 1¼×C = 1¼ × 4 = 5 and ABCDEF = 684365