Puzzle for December 25, 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
eq.6 may be written as: A + F + C = B + D + E In the above equation, replace A + F with B + E (from eq.4): B + E + C = B + D + E Subtract B and E from both sides: B + E + C - B - E = B + D + E - B - E which simplifies to C = D
Hint #2
In eq.5, replace D with C: F - C = C + C which becomes F - C = 2×C Add C to both sides of the above equation: F - C + C = 2×C + C which makes F = 3×C
Hint #3
In eq.3, substitute 3×C for F: E - 3×C = 3×C - C which becomes E - 3×C = 2×C Add 3×C to both sides of the above equation: E - 3×C + 3×C = 2×C + 3×C which makes E = 5×C
Hint #4
Substitute 5×C for E in eq.2: B = C + 5×C which makes B = 6×C
Hint #5
Substitute 3×C for F, 6×C for B, and 5×C for E in eq.4: A + 3×C = 6×C + 5×C which becomes A + 3×C = 11×C Subtract 3×C from each side of the equation above: A + 3×C - 3×C = 11×C - 3×C which makes A = 8×C
Solution
Substitute 8×C for A, 6×C for B, C for D, 5×C for E, and 3×C for F in eq.1: 8×C + 6×C + C + C + 5×C + 3×C = 24 which simplifies to 24×C = 24 Divide both sides of the above equation by 24: 24×C ÷ 24 = 24 ÷ 24 which means C = 1 making A = 8×C = 8 × 1 = 8 B = 6×C = 6 × 1 = 6 D = C = 1 E = 5×C = 5 × 1 = 5 F = 3×C = 3 × 1 = 3 and ABCDEF = 861153