Puzzle for November 16, 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, replace C with E + F (from eq.2): A + F = E + F + E which becomes A + F = 2×E + F Subtract F from each side of the equation above: A + F - F = 2×E + F - F which makes A = 2×E
Hint #2
In eq.3, replace A with 2×E: eq.3a) C = 2×E + B
Hint #3
In eq.5, substitute A + B for C (from eq.3): A + F = A + B + E Subtract A from each side of the above equation: A + F - A = A + B + E - A which becomes eq.5a) F = B + E
Hint #4
Substitute 2×E + B for C (from eq.3a), 2×E for A, and B + E for F (from eq.5a) in eq.6: B + 2×E + B = 2×E + D + E + B + E which becomes 2×B + 2×E = 4×E + D + B Subtract 4×E and B from both sides of the equation above: 2×B + 2×E - 4×E - B = 4×E + D + B - 4×E - B which becomes eq.6a) B - 2×E = D
Hint #5
Substitute B - 2×E for D (from eq.6a), and 2×E + B for C (from eq.3a) in eq.4: B + B - 2×E = 2×E + B + E which becomes 2×B - 2×E = 3×E + B In the above equation, add 2×E to both sides, and subtract B from both sides: 2×B - 2×E + 2×E - B = 3×E + B + 2×E - B which simplifies to B = 5×E
Hint #6
Substitute 5×E for B in eq.6a: 5×E - 2×E = D which makes 3×E = D
Hint #7
Substitute 5×E for B in eq.5a: F = 5×E + E which makes F = 6×E
Hint #8
Substitute 5×E for B in eq.3a: C = 2×E + 5×E which makes C = 7×E
Solution
Substitute 2×E for A, 5×E for B, 7×E for C, 3×E for D, and 6×E for F in eq.1: 2×E + 5×E + 7×E + 3×E + E + 6×E = 24 which simplifies to 24×E = 24 Divide both sides of the above equation by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = 2×E = 2 × 1 = 2 B = 5×E = 5 × 1 = 5 C = 7×E = 7 × 1 = 7 D = 3×E = 3 × 1 = 3 F = 6×E = 6 × 1 = 6 and ABCDEF = 257316