Puzzle for November 15, 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.6, replace B with D + F (from eq.3), and A with E + F (from eq.2): D + F + F = E + F + C + D + E which becomes D + 2×F = 2×E + F + C + D Subtract D and F from each side of the above equation: D + 2×F - D - F = 2×E + F + C + D - D - F which becomes eq.6a) F = 2×E + C
Hint #2
In eq.2, replace F with 2×E + C (from eq.6a): A = E + C + 2×E which becomes eq.2a) A = 3×E + C
Hint #3
In eq.4, substitute 3×E + C for A (from eq.2a): C + D = 3×E + C + E which becomes C + D = 4×E + C Subtract C from each side of the equation above: C + D - C = 4×E + C - C which makes D = 4×E
Hint #4
Substitute 3×E + C for A (from eq.2a), and 4×E for D in eq.5: 3×E + C + 4×E = B + C which becomes 7×E + C = B + C Subtract C from both sides of the above equation: 7×E + C - C = B + C - C which makes 7×E = B
Hint #5
Substitute 7×E for B, and 4×E for D in eq.3: 7×E = 4×E + F Subtract 4×E from each side of the above equation: 7×E - 4×E = 4×E + F - 4×E which makes 3×E = F
Hint #6
Substitute 3×E for F in eq.6a: 3×E = 2×E + C Subtract 2×E from both sides of the equation above: 3×E - 2×E = 2×E + C - 2×E which makes E = C
Hint #7
Substitute E for C in eq.2a: A = 3×E + E which makes A = 4×E
Solution
Substitute 4×E for A and D, 7×E for B, E for C, and 3×E for F in eq.1: 4×E + 7×E + E + 4×E + E + 3×E = 20 which simplifies to 20×E = 20 Divide both sides of the above equation by 20: 20×E ÷ 20 = 20 ÷ 20 which means E = 1 making A = D = 4×E = 4 × 1 = 4 B = 7×E = 7 × 1 = 7 C = E = 1 F = 3×E = 3 × 1 = 3 and ABCDEF = 471413