Puzzle for November 17, 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.5, replace A with D + E (from eq.2): E + F = D + E – F In the above equation, subtract E from both sides, and add F to both sides: E + F – E + F = D + E – F – E + F which simplifies to 2×F = D
Hint #2
In eq.4, replace D with 2×F: C + 2×F = B – 2×F Add 2×F to both sides of the above equation: C + 2×F + 2×F = B – 2×F + 2×F which becomes eq.4a) C + 4×F = B
Hint #3
In eq.3, substitute C + 4×F for B (from eq.4a): A + C = C + 4×F Subtract C from each side of the above equation: A = 4×F
Hint #4
Substitute 2×F for D, and 4×F for A in eq.2: 2×F + E = 4×F Subtract 2×F from each side of the equation above: 2×F + E – 2×F = 4×F – 2×F which makes E = 2×F
Hint #5
Substitute 4×F for A, and 2×F for D and E in eq.6: 4×F + F = B + C + 2×F + 2×F – 4×F which becomes eq.6a) 5×F = B + C
Hint #6
Substitute C + 4×F for B (from eq.4a) into eq.6a: 5×F = C + 4×F + C which becomes 5×F = 2×C + 4×F Subtract 4×F from each side of the equation above: 5×F – 4×F = 2×C + 4×F – 4×F which makes F = 2×C Divide both sides by 2: F ÷ 2 = 2×C ÷ 2 which makes ½×F = C
Hint #7
Substitute ½×F for C in eq.4a: ½×F + 4×F = B which makes 4½×F = B
Solution
Substitute 4×F for A, 4½×F for B, ½×F for C, and 2×F for D and E in eq.1: 4×F + 4½×F + ½×F + 2×F + 2×F + F = 28 which simplifies to 14×F = 28 Divide both sides of the above equation by 14: 14×F ÷ 14 = 28 ÷ 14 which means F = 2 making A = 4×F = 4 × 2 = 8 B = 4½×F = 4½ × 2 = 9 C = ½×F = ½ × 2 = 1 D = E = 2×F = 2 × 2 = 4 and ABCDEF = 891442