Puzzle for December 26, 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.
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Hint #1
Add C and E to both sides of eq.4: A - C + C + E = C - E + C + E which becomes eq.4a) A + E = 2×C
Hint #2
In eq.6, replace A + E with 2×C (from eq.4a), and D with C + F (from eq.3): 2×C = C + F + F which becomes 2×C = C + 2×F Subtract C from each side of the equation above: 2×C - C = C + 2×F - C which makes eq.6a) C = 2×F
Hint #3
In eq.5, substitute 2×F for C: 2×F - A = A + F In the above equation, add A to both sides, and subtract F from both sides: 2×F - A + A - F = A + F + A - F which makes F = 2×A
Hint #4
Substitute (2×A) for F in eq.6a: C = 2×(2×A) which makes C = 4×A
Hint #5
Substitute (4×A) for C in eq.4a: A + E = 2×(4×A) which becomes A + E = 8×A Subtract A from each side of the above equation: A + E - A = 8×A - A which makes E = 7×A
Hint #6
Substitute 4×A for C, and 2×A for F in eq.3: D = 4×A + 2×A which makes D = 6×A
Hint #7
Substitute 7×A for E, and 2×A for F in eq.2: B = 7×A + 2×A which makes B = 9×A
Solution
Substitute 9×A for B, 4×A for C, 6×A for D, 7×A for E, and 2×A for F in eq.1: A + 9×A + 4×A + 6×A + 7×A + 2×A = 29 which simplifies to 29×A = 29 Divide both sides of the above equation by 29: 29×A ÷ 29 = 29 ÷ 29 which means A = 1 making B = 9×A = 9 × 1 = 9 C = 4×A = 4 × 1 = 4 D = 6×A = 6 × 1 = 6 E = 7×A = 7 × 1 = 7 F = 2×A = 2 × 1 = 2 and ABCDEF = 194672