Puzzle for May 17, 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 CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
Add D to both sides of eq.2: F - D + D = A + B + D which becomes eq.2a) F = A + B + D Subtract A from both sides of eq.2a: F - A = A + B + D - A which becomes eq.2b) F - A = B + D
Hint #2
In eq.5, replace F with A + B + D (from eq.2a): B + A + B + D = A + C + D which becomes 2×B + A + D = A + C + D Subtract both A and D from each side of the equation above: 2×B + A + D - A - D = A + C + D - A - D which simplifies to 2×B = C
Hint #3
In eq.4, replace C with 2×B: E - A + B = 2×B + D Subtract B from both sides of the above equation: E - A + B - B = 2×B + D - B which becomes eq.4a) E - A = B + D
Hint #4
In eq.2b, substitute E - A for B + D (from eq.4a): F - A = E - A Add A to both sides: F - A + A = E - A + A which makes F = E
Hint #5
Substitute E for F in eq.3: D + E = C + E Subtract E from each side of the above equation: D + E - E = C + E - E which makes D = C and which means D = C = 2×B
Hint #6
eq.6 may be written as: 10×C + D = 10×A + B + A + E which becomes 10×C + D = 11×A + B + E Substitute 2×B for C and D in the above equation: 10×2×B + 2×B = 11×A + B + E which becomes 22×B = 11×A + B + E Subtract B and 11×A from each side: 22×B - B - 11×A = 11×A + B + E - B - 11×A which becomes eq.6a) 21×B - 11×A = E
Hint #7
Substitute 21×B - 11×A for E (from eq.6a), and 2×B for C and D in eq.4: 21×B - 11×A - A + B = 2×B + 2×B which becomes 22×B - 12×A = 4×B In the above equation, add 12×A to both sides, and subtract 4×B from both sides: 22×B - 12×A + 12×A - 4×B = 4×B + 12×A - 4×B which becomes 18×B = 12×A Divide both sides by 12: 18×B ÷ 12 = 12×A ÷ 12 which makes 1½×B = A
Hint #8
Substitute (1½×B) for A in eq.6a: 21×B - 11×(1½×B) = E which is the same as 21×B - 16½×B = E which makes 4½×B = E and which also makes F = E = 4½×B
Solution
Substitute 1½×B for A, 2×B for C and D, and 4½×B for E and F in eq.1: 1½×B + B + 2×B + 2×B + 4½×B + 4½×B = 31 which simplifies to 15½×B = 31 Divide both sides of the above equation by 15½: 15½×B ÷ 15½ = 31 ÷ 15½ which means B = 2 making A = 1½×B = 1½ × 2 = 3 C = D = 2×B = 2 × 2 = 4 E = F = 4½×B = 4½ × 2 = 9 and ABCDEF = 324499