Puzzle for August 17, 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
In eq.5, subtract E from both sides, and add F to both sides: B - D + E - E + F = C + D - F - E + F which becomes eq.5a) B - D + F = C + D - E
Hint #2
In eq.6, replace B - D + F with C + D - E (from eq.5a): A + D - F = C + D - E Subtract D from both sides of the above equation: A + D - F - D = C + D - E - D which becomes A - F = C - E Add F and E to both sides: A - F + F + E = C - E + F + E which becomes eq.6a) A + E = C + F
Hint #3
In eq.4, substitute C + F for A + E (from eq.6a): C + F = C + F - F which becomes C + F = C Subtract C from each side of the equation above: C + F - C = C - C which makes F = 0
Hint #4
Substitute 0 for F in eq.2: B + D = C + 0 which becomes eq.2a) B + D = C
Hint #5
In eq.5, substitute B + D for C (from eq.2a), and 0 for F: B - D + E = B + D + D - 0 which becomes B - D + E = B + 2×D In the above equation, subtract B from both sides, and add D to both sides: B - D + E - B + D = B + 2×D - B + D which simplifies to E = 3×D
Hint #6
Substitute 0 for F, and 3×D for E in eq.3: A + 0 = D + 3×D which makes A = 4×D
Hint #7
Substitute 4×D for A, 3×D for E, and 0 for F in eq.6a: 4×D + 3×D = C + 0 which makes 7×D = C
Hint #8
Substitute 7×D for C in eq.2a: B + D = 7×D Subtract D from each side of the above equation: B + D - D = 7×D - D which makes B = 6×D
Solution
Substitute 4×D for A, 6×D for B, 7×D for C, 3×D for E, and 0 for F in eq.1: 4×D + 6×D + 7×D + D + 3×D + 0 = 21 which simplifies to 21×D = 21 Divide both sides of the above equation by 21: 21×D ÷ 21 = 21 ÷ 21 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 6×D = 6 × 1 = 6 C = 7×D = 7 × 1 = 7 E = 3×D = 3 × 1 = 3 and ABCDEF = 467130