Puzzle for July 11, 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 = B + E + C In the above equation, replace B + E with A + C (from eq.4): A + F = A + C + C which becomes A + F = A + 2×C Subtract A from each side of the above equation: A + F - A = A + 2×C - A which makes F = 2×C
Hint #2
In eq.2, replace F with 2×C: 2×C = C + D Subtract C from each side of the above equation: 2×C - C = C + D - C which makes D = C
Hint #3
In eq.3, substitute C for D: C + E = B + C Subtract C from both sides of the equation above: C + E - C = B + C - C which makes E = B
Hint #4
In eq.5, replace F with 2×C: 2×C - E = A - 2×C Add 2×C to both sides of the above equation: 2×C - E + 2×C = A - 2×C + 2×C which becomes eq.5a) 4×C - E = A
Hint #5
Substitute 4×C - E for A (from eq.5a), 2×C for F, and E for B in eq.6: 4×C - E + 2×C = E + C + E which becomes 6×C - E = 2×E + C In the above equation, add E to both sides, and subtract C from both sides: 6×C - E + E - C = 2×E + C + E - C which becomes 5×C = 3×E Divide both sides of the above equation by 3: 5×C ÷ 3 = 3×E ÷ 3 which makes 1⅔×C = E and also makes 1⅔×C = E = B
Hint #6
Substitute 1⅔×C for E in eq.5a: 4×C - 1⅔×C = A which makes 2⅓×C = A
Solution
Substitute 2⅓×C for A, 1⅔×C for B and E, C for D, and 2×C for F in eq.1: 2⅓×C + 1⅔×C + C + C + 1⅔×C + 2×C = 29 which simplifies to 9⅔×C = 29 Divide both sides of the above equation by 9⅔: 9⅔×C ÷ 9⅔ = 29 ÷ 9⅔ which means C = 3 making A = 2⅓×C = 2⅓ × 3 = 7 B = E = 1⅔×C = 1⅔ × 3 = 5 D = C = 3 F = 2×C = 2 × 3 = 6 and ABCDEF = 753356