Puzzle for January 3, 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.
* EF is a 2-digit number (not E×F).
Scratchpad
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Hint #1
In eq.2, replace D with A + B (from eq.5): A + F = A + B + E Subtract A from both sides of the equation above: A + F - A = A + B + E - A which becomes eq.2a) F = B + E
Hint #2
In eq.3, replace F with B + E (from eq.2a): D + B + E = A + C + E Subtract E from both sides: D + B + E - E = A + C + E - E which becomes eq.3a) D + B = A + C
Hint #3
Substitute A + B for D (from eq.5) in eq.3a: A + B + B = A + C Subtract A from each side: A + B + B - A = A + C - A which makes 2×B = C
Hint #4
Substitute 2×B for C in eq.4: B + 2×B - F = F Add F to both sides: B + 2×B - F + F = F + F which simplifies to 3×B = 2×F Divide both sides by 2: 3×B ÷ 2 = 2×F ÷ 2 which makes 1½×B = F
Hint #5
In eq.2a, replace F with 1½×B: 1½×B = B + E Subtract B from each side of the above equation: 1½×B - B = B + E - B which becomes ½×B = E
Hint #6
eq.6 may be written as: B + D = 10×E + F - B Subtract B from both sides of the above equation: B + D - B = 10×E + F - B - B which becomes D = 10×E + F - 2×B Substitute (½×B) for E, and 1½×B for F: D = 10×(½×B) + 1½×B - 2×B which becomes D = 5×B + 1½×B - 2×B which makes D = 4½×B
Hint #7
Substitute 4½×B for D in eq.5: A + B = 4½×B Subract B from each side: A + B - B = 4½×B - B which means A = 3½×B
Solution
Substitute 3½×B for A, 2×B for C, 4½×B for D, ½×B for E, and 1½×B for F in eq.1: 3½×B + B + 2×B + 4½×B + ½×B + 1½×B = 26 which simplifies to 13×B = 26 Divide both sides by 13: 13×B ÷ 13 = 26 ÷ 13 which makes B = 2 making A = 3½×B = 3½ × 2 = 7 C = 2×B = 2 × 2 = 4 D = 4½×B = 4½ × 2 = 9 E = ½×B = ½ × 2 = 1 F = 1½×B = 1½ × 2 = 3 and ABCDEF = 724913