Puzzle for August 5, 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.
* EF and BC are 2-digit numbers (not E×F or B×C).
Scratchpad
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Hint #1
eq.4 may be written as: eq.4a) A + 10×E + F = 10×B + C + D + E
Hint #2
In eq.4a, replace C + D + E with A + F (from eq.3): A + 10×E + F = 10×B + A + F Subtract A and F from each side of the above equation: A + 10×E + F - A - F = 10×B + A + F - A - F which simplifies to 10×E = 10×B Divide both sides by 10: 10×E ÷ 10 = 10×B ÷ 10 which makes E = B
Hint #3
In eq.5, replace E with B: F - A = B ÷ B which becomes F - A = 1 Add A to both sides of the equation above: F - A + A = 1 + A which becomes eq.5a) F = 1 + A
Hint #4
In eq.2, substitute (1 + A) for F (from eq.5a): A - (1 + A) = C - D which becomes A - 1 - A = C - D which becomes -1 = C - D Add 1 and D to both sides of the above equation: -1 + 1 + D = C - D + 1 + D which becomes eq.2a) D = C + 1
Hint #5
In eq.6, substitute B for E, (1 + A) for F (from eq.5a), and C + 1 for D (from eq.2a): B × (1 + A) = (A × B) + C + 1 which becomes B + (B × A) = (A × B) + C + 1 which is equivalent to B + (A × B) = (A × B) + C + 1 Subtract (A × B) from each side of the equation above: B + (A × B) - (A × B) = (A × B) + C + 1 - (A × B) which makes B = C + 1 and also makes eq.6a) E = B = D = C + 1
Hint #6
Substitute 1 + A for F (from eq.5a), and C + 1 for D and E (from eq.6a) in eq.3: A + 1 + A = C + C + 1 + C + 1 which becomes 2×A + 1 = 3×C + 2 Subtract 1 from each side of the above equation: 2×A + 1 - 1 = 3×C + 2 - 1 which becomes 2×A = 3×C + 1 Divide both sides by 2: 2×A ÷ 2 = (3×C + 1) ÷ 2 which makes eq.3a) A = 1½×C + ½
Hint #7
Substitute 1½×C + ½ for A (from eq.3a) into eq.5a: F = 1 + 1½×C + ½ which becomes eq.5b) F = 1½ + 1½×C
Hint #8
Substitute 1½×C + ½ for A (from eq.3a), C + 1 for B and D and E (from eq.6a), and 1½ + 1½×C for F (from eq.5b) in eq.1: 1½×C + ½ + C + 1 + C + C + 1 + C + 1 + 1½ + 1½×C = 40 which simplifies to 7×C + 5 = 40 Subtract 5 from both sides of the above equation: 7×C + 5 - 5 = 40 - 5 which makes 7×C = 35 Divide both sides by 7: 7×C ÷ 7 = 35 ÷ 7 which means C = 5
Solution
Since C = 5, then: A = 1½×C + ½ = 1½×5 + ½ = 7½ + ½ = 8 (from eq.3a) B = D = E = C + 1 = 5 + 1 = 6 (from eq.6a) F = 1½ + 1½×C = 1½ + 1½×5 = 1½ + 7½ = 9 (from eq.5b) and ABCDEF = 865669