Puzzle for December 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.
Scratchpad
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Hint #1
In eq.2, replace F with B + D + E (from eq.4): A = B + B + D + E which becomes eq.2a) A = 2×B + D + E
Hint #2
In eq.6, replace A with 2×B + D + E (from eq.2a), and F with B + D + E (from eq.4): 2×B + D + E + E = B + D + B + D + E which becomes 2×B + D + 2×E = 2×B + 2×D + E Subtract 2×B, D, and E from each side of the equation above: 2×B + D + 2×E - 2×B - D - E = 2×B + 2×D + E - 2×B - D - E which simplifies to E = D
Hint #3
In eq.4, substitute D for E: F = B + D + D which becomes eq.4a) F = B + 2×D
Hint #4
In eq.2a, substitute D for E: A = 2×B + D + D which becomes eq.2b) A = 2×B + 2×D
Hint #5
In eq.3, replace A with 2×B + 2×D (from eq.2b): C = 2×B + 2×D + D which becomes eq.3a) C = 2×B + 3×D
Hint #6
Substitute B + 2×D for F (from eq.4a), (2×B + 2×D) for A (from eq.2b), and (2×B + 3×D) for C (from eq.3a) in eq.5: B + 2×D - (2×B + 2×D) = (2×B + 2×D) - (2×B + 3×D) which becomes B + 2×D - 2×B - 2×D = 2×B + 2×D - 2×B - 3×D which simplifies to -B = -D which means B = D and also means B = D = E
Hint #7
Substitute D for B in eq.3a: C = 2×D + 3×D which makes C = 5×D
Hint #8
Substitute D for B in eq.2b: A = 2×D + 2×D which makes A = 4×D
Hint #9
Substitute D for B in eq.4a: F = D + 2×D which makes F = 3×D
Solution
Substitute 4×D for A, D for B and E, 5×D for C, and 3×D for F in eq.1: 4×D + D + 5×D + D + D + 3×D = 15 which simplifies to 15×D = 15 Divide both sides of the above equation by 15: 15×D ÷ 15 = 15 ÷ 15 which means D = 1 making A = 4×D = 4 × 1 = 4 B = E = D = 1 C = 5×D = 5 × 1 = 5 F = 3×D = 3 × 1 = 3 and ABCDEF = 415113