Puzzle for November 10, 2022 ( )
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 to both sides of eq.2: B + C + E + F = A + D - F + F which becomes eq.2a) B + C + E + F = A + D which may be written as eq.2b) B + F + C + E = A + D
Hint #2
In eq.2b, replace B + F with C + D (from eq.3): C + D + C + E = A + D which becomes 2×C + D + E = A + D Subtract D from each side of the equation above: 2×C + D + E - D = A + D - D which becomes eq.2c) 2×C + E = A
Hint #3
In eq.4, replace A with 2×C + E (from eq.2c): D + E = 2×C + E + C which becomes D + E = 3×C + E Subtract E from each side of the above equation: D + E - E = 3×C + E - E which makes eq.4a) D = 3×C
Hint #4
In eq.2a, substitute A + B + C for E + F (from eq.5), and 3×C for D: B + C + A + B + C = A + 3×C which becomes 2×B + 2×C + A = A + 3×C Subtract 2×C and A from both sides of the above equation: 2×B + 2×C + A - 2×C - A = A + 3×C - 2×C - A which simplifies to 2×B = C
Hint #5
Substitute (2×B) for C in eq.4a: D = 3×(2×B) which makes D = 6×B
Hint #6
Substitute 2×B for C, and 6×B for D in eq.3: 2×B + 6×B = B + F which becomes 8×B = B + F Subtract B from each side of the equation above: 8×B - B = B + F - B which makes 7×B = F
Hint #7
Substitute 7×B for F in eq.6: 7×B - E = A - 7×B Add 7×B to both sides of the above equation: 7×B - E + 7×B = A - 7×B + 7×B which becomes eq.6a) 14×B - E = A
Hint #8
Substitute 6×B for D, 14×B - E for A (from eq.6a), and 2×B for C in eq.4: 6×B + E = 14×B - E + 2×B which becomes 6×B + E = 16×B - E In the above equation, subtract 6×B from both sides, and add E to both sides: 6×B + E - 6×B + E = 16×B - E - 6×B + E which makes 2×E = 10×B Divide both sides by 2: 2×E ÷ 2 = 10×B ÷ 2 which makes E = 5×B
Hint #9
Substitute 5×B for E in eq.6a: 14×B - 5×B = A which makes 9×B = A
Solution
Substitute 9×B for A, 2×B for C, 6×B for D, 5×B for E, and 7×B for F in eq.1: 9×B + B + 2×B + 6×B + 5×B + 7×B = 30 which simplifies to 30×B = 30 Divide both sides of the above equation by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = 9×B = 9 × 1 = 9 C = 2×B = 2 × 1 = 2 D = 6×B = 6 × 1 = 6 E = 5×B = 5 × 1 = 5 F = 7×B = 7 × 1 = 7 and ABCDEF = 912657