Puzzle for October 20, 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.
* DE is a 2-digit number (not D×E).
Scratchpad
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Hint #1
Subtract the left and right sides of eq.4 from the left and right sides of eq.3, respectively: A + B - (B + C) = C + D + E - (A + D + E) which becomes A + B - B - C = C + D + E - A - D - E which becomes A - C = C - A Add C and A to both sides of the above equation: A - C + C + A = C - A + C + A which becomes 2×A = 2×C Divide both sides by 2: 2×A ÷ 2 = 2×C ÷ 2 which makes A = C
Hint #2
In eq.5, replace B + D with F (from eq.2): A - D - F = F + F which becomes A - D - F = 2×F Add D and F to both sides of the equation above: A - D - F + D + F = 2×F + D + F which makes A = 3×F + D and also makes eq.5a) A = C = 3×F + D
Hint #3
Subtract D from each side of eq.2: F - D = B + D - D which becomes eq.2a) F - D = B
Hint #4
In eq.3, replace A and C with 3×F + D (from eq.5a), and B with F - D (from eq.2a): 3×F + D + F - D = 3×F + D + D + E which becomes 4×F = 3×F + 2×D + E Subtract 3×F and 2×D from each side of the equation above: 4×F - 3×F - 2×D = 3×F + 2×D + E - 3×F - 2×D which becomes eq.3a) F - 2×D = E
Hint #5
eq.6 may be written as: 10×D + E = B + C + F In the above equation, substitute F - 2×D for E (from eq.3a), F - D for B (from eq.2a), and 3×F + D for C (from eq.5a): 10×D + F - 2×D = F - D + 3×F + D + F which becomes 8×D + F = 5×F Subtract F from each side of the above equation: 8×D + F - F = 5×F - F which makes 8×D = 4×F Divide both sides by 4: 8×D ÷ 4 = 4×F ÷ 4 which makes 2×D = F
Hint #6
Substitute (2×D) for F in eq.5a: A = C = 3×(2×D) + D which becomes A = C = 6×D + D which makes A = C = 7×D
Hint #7
Substitute 2×D for F in eq.2a: 2×D - D = B which makes D = B
Hint #8
Substitute 2×D for F in eq.3a: 2×D - 2×D = E which makes 0 = E
Solution
Substitute 7×D for A and C, D for B, 0 for E, and 2×D for F in eq.1: 7×D + D + 7×D + D + 0 + 2×D = 18 which simplifies to 18×D = 18 Divide both sides of the above equation by 18: 18×D ÷ 18 = 18 ÷ 18 which means D = 1 making A = C = 7×D = 7×1 = 7 B = D = 1 F = 2×D = 2×1 = 2 making ABCDEF = 717102