Puzzle for February 12, 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 DE are 2-digit numbers (not A×B or D×E).
Scratchpad
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Hint #1
Add C + D to both sides of eq.4: B - C + C + D = A - D + F + C + D which becomes B + D = A + F + C Subtract the left and right sides of eq.2 from the left and right sides of the above equation, respectively: B + D - B = A + C + F - (A + C) which is the same as B + D - B = A + C + F - A - C which makes D = F
Hint #2
In eq.3, replace F with D: D + D = E which makes 2×D = E
Hint #3
In eq.5, replace E with 2×D, and F with D: C = 2×D + D which makes C = 3×D
Hint #4
eq.6 may be written as: 10×A + B = B + C + 10×D + E Subtract B from both sides of the above equation: 10×A + B - B = B + C + 10×D + E - B which becomes 10×A = C + 10×D + E Substitute 3×D for C, and 2×D for E: 10×A = 3×D + 10×D + 2×D which becomes 10×A = 15×D Divide both sides by 10: 10×A ÷ 10 = 15×D ÷ 10 which makes A = 1½×D
Hint #5
Substitute 1½×D for A, and 3×D for C in eq.2: B = 1½×D + 3×D which means B = 4½×D
Solution
Substitute 1½×D for A, 4½×D for B, 3×D for C, 2×D for E, and D for F in eq.1: 1½×D + 4½×D + 3×D + D + 2×D + D = 26 which becomes 13×D = 26 Divide both sides of the equation above by 13: 13×D ÷ 13 = 26 ÷ 13 which makes D = 2 making A = 1½×D = 1½ × 2 = 3 B = 4½×D = 4½ × 2 = 9 C = 3×D = 3 × 2 = 6 E = 2×D = 2 × 2 = 4 F = D = 2 and ABCDEF = 396242