Puzzle for November 25, 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.
Scratchpad
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Hint #1
In eq.2, replace D with C + E (from eq.5): B + C = C + E – B In the equation above, subtract C from both sides, and add B to both sides: B + C – C + B = C + E – B – C + B which simplifies to 2×B = E
Hint #2
In eq.4, substitute 2×B for E: 2×B = B + F Subtract B from both sides of the above equation: 2×B – B = B + F – B which makes B = F
Hint #3
Substitute C – B for A (from eq.3), 2×B for E, and B for F in eq.6: C – B + C = B + 2×B + B which becomes 2×C – B = 4×B Add B to both sides of the above equation: 2×C – B + B = 4×B + B which makes 2×C = 5×B Divide both sides by 2: 2×C ÷ 2 = 5×B ÷ 2 which makes C = 2½×B
Hint #4
Substitute 2½×B for C, and 2×B for E in eq.5: D = 2½×B + 2×B which makes D = 4½×B
Hint #5
Substitute 2½×B for C in eq.3: 2½×B – B = A which makes 1½×B = A
Solution
Substitute 1½×B for A, 2½×B for C, 4½×B for D, 2×B for E, and B for F in eq.1: 1½×B + B + 2½×B + 4½×B + 2×B + B = 25 which simplifies to 12½×B = 25 Divide both sides of the equation above by 12½: 12½×B ÷ 12½ = 25 ÷ 12½ which means B = 2 making A = 1½×B = 1½ × 2 = 3 C = 2½×B = 2½ × 2 = 5 D = 4½×B = 4½ × 2 = 9 E = 2×B = 2 × 2 = 4 F = B = 2 and ABCDEF = 325942