Puzzle for February 2, 2019 ( )
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.
* AB and BC are 2-digit numbers (not A×B or B×C).
Scratchpad
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Hint #1
Add B + E to both sides of eq.5: A - B + B + E = D - E + B + E which becomes A + E = D + B In the above equation, replace D + B wtith A + C (from eq.3): A + E = A + C Subtract A from each side: A + E - A = A + C - A which makes E = C
Hint #2
In eq.2, replace E with C: C = C + F Subtract C from each side of the above equation: C - C = C + F - C which makes 0 = F
Hint #3
In eq.4, replace E with C: C + C - A = A - B + D which becomes 2×C - A = A - B + D Add (B - A) to each side of the above equation: 2×C - A + (B - A) = A - B + D + (B - A) which becomes eq.4a) 2×C + B - 2×A = D
Hint #4
Subtract B from both sides of eq.3: A + C - B = B + D - B which becomes eq.3a) A + C - B = D In eq.4a, substitute A + C - B for D: 2×C + B - 2×A = A + C - B Subtract C from each side of the above equation: 2×C + B - 2×A - C = A + C - B - C which becomes C + B - 2×A = A - B Add (2×A - B) to each side: C + B - 2×A + (2×A - B) = A - B + (2×A - B) which simplifies to eq.4b) C = 3×A - 2×B
Hint #5
eq.6 may be written as: 10×A + B - (10×B + C) = A + D - E which is the same as 10×A + B - 10×B - C = A + D - E In the above equation, substitute A + C - B for D (from eq.3a), and C for E: 10×A + B - 10×B - C = A + A + C - B - C which becomes eq.6a) 10×A - 9×B - C = 2×A - B
Hint #6
In eq.6a, substitute (3×A - 2×B) for C (from eq.4b): 10×A - 9×B - (3×A - 2×B) = 2×A - B which is the same as 10×A - 9×B - 3×A + 2×B = 2×A - B which becomes 7×A - 7×B = 2×A - B Add (7×B - 2×A) to both sides of the equation above: 7×A - 7×B + (7×B - 2×A) = 2×A - B + (7×B - 2×A) which simplifies to 5×A = 6×B Divide both sides by 5: 5×A ÷ 5 = 6×B ÷ 5 which makes A = 1⅕×B
Hint #7
Substitute (1⅕×B) for A in eq.4b: C = 3×(1⅕×B) - 2×B which becomes C = 3⅗×B - 2×B which makes C = 1⅗×B
Hint #8
Substitute (1⅗×B) for C, and (1⅕×B) for A in eq.4a: 2×(1⅗×B) + B - 2×(1⅕×B) = D which becomes 3⅕×B + B - 2⅖×B = D which makes 1⅘×B = D
Solution
Substitute 1⅕×B for A, 1⅗×B for C and E, 1⅘×B for D, and 0 for F in eq.1: 1⅕×B + B + 1⅗×B + 1⅘×B + 1⅗×B + 0 = 36 which simplifies to 7⅕×B = 36 Divide both sides of the equation above by 7⅕: 7⅕×B ÷ 7⅕ = 36 ÷ 7⅕ which makes B = 5 making A = 1⅕×B = 1⅕ × 5 = 6 C = E = 1⅗×B = 1⅗ × 5 = 8 D = 1⅘×B = 1⅘ × 5 = 9 and ABCDEF = 658980