Puzzle for September 8, 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.
* BC is a 2-digit number (not B×C).
Scratchpad
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Hint #1
Add B, C, and F to both sides of eq.3: A - B + B + C + F = E - C - F + B + C + F which becomes A + C + F = E + B which may be written as eq.3a) A + C + F = B + E
Hint #2
In eq.4, replace B + E with A + C + F (from eq.3a): A + C + F + F = A + D - F which becomes A + C + 2×F = A + D - F In the above equation, subtract A from both sides, and add F to both sides: A + C + 2×F - A + F = A + D - F - A + F which becomes eq.4a) C + 3×F = D
Hint #3
In eq.5, replace D with C + 3×F (from eq.4a): C + 3×F + C - F = B - C + F which becomes 2×C + 2×F = B - C + F In the equation above, add C to both sides, and subtract F from both sides: 2×C + 2×F + C - F = B - C + F + C - F which becomes eq.5a) 3×C + F = B
Hint #4
In eq.2, substitute 3×C + F for B (from eq.5a): E + F = A + 3×C + F Subtract F from each side of the equation above: E + F - F = A + 3×C + F - F which becomes eq.2a) E = A + 3×C
Hint #5
Substitute 3×C + F for B (from eq.5a), A + 3×C for E (from eq.2a), and C + 3×F for D (from eq.4a) into eq.4: 3×C + F + A + 3×C + F = A + C + 3×F - F which becomes 6×C + 2×F + A = A + C + 2×F Subtract A, C, and 2×F from each side of the above equation: 6×C + 2×F + A - A - C - 2×F = A + C + 2×F - A - C - 2×F which simplifies to 5×C = 0 which means C = 0
Hint #6
Substitute 0 for C in eq.2a: E = A + 3×0 which becomes E = A + 0 which makes E = A
Hint #7
Substitute 0 for C in eq.5a: 3×0 + F = B which becomes 0 + F = B which makes F = B
Hint #8
Substitute 0 for C, and B for F in eq.4a: 0 + 3×B = D which makes 3×B = D
Hint #9
eq.6 may be written as eq.6a) 10×B + C = A + E + F
Hint #10
Substitute 0 for C, A for E, and B for F in eq.6a: 10×B + 0 = A + A + B which becomes 10×B = 2×A + B Subtract B from both sides of the equation above: 10×B - B = 2×A + B - B which makes 9×B = 2×A Divide both sides by 2: 9×B ÷ 2 = 2×A ÷ 2 which makes 4½×B = A and also makes 4½×B = A = E
Solution
Substitute 4½×B for A and E, 0 for C, 3×B for D, and B for F in eq.1: 4½×B + B + 0 + 3×B + 4½×B + B = 28 which simplifies to 14×B = 28 Divide both sides of the above equation by 14: 14×B ÷ 14 = 28 ÷ 14 which means B = 2 making A = E = 4½×B = 4½ × 2 = 9 D = 3×B = 3 × 2 = 6 F = B = 2 and ABCDEF = 920692