Puzzle for February 17, 2020 ( )
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.5, replace E with B + C (from eq.4): C + D = B + B + C which becomes C + D = 2×B + C Subtract C from both sides of the above equation: C + D – C = 2×B + C – C which makes D = 2×B
Hint #2
In eq.3, replace D with 2×B: 2×B = A + B Subtract B from both sides of the above equation: 2×B – B = A + B – B which makes B = A
Hint #3
In eq.2, substitute B for A: B + F = B + C Subtract B from both sides of the above equation: B + F – B = B + C – B which makes F = C
Hint #4
Substitute B for A, 2×B for D, and B + C for E (from eq.4) in eq.6: B + B + 2×B = C + B + C – 2×B which becomes 4×B = 2×C – B Add B to both sides of the above equation: 4×B + B = 2×C – B + B which means 5×B = 2×C Divide both sides by 2: 5×B ÷ 2 = 2×C ÷ 2 which makes 2½×B = C and also makes F = C = 2½×B
Hint #5
Substitute 2½×B for C in eq.4: E = B + 2½×B which makes E = 3½×B
Solution
Substitute B for A, 2½×B for C and F, 2×B for D, and 3½×B for E in eq.1: B + B + 2½×B + 2×B + 3½×B + 2½×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 = B = 2 C = F = 2½×B = 2½ × 2 = 5 D = 2×B = 2 × 2 = 4 E = 3½×B = 3½ × 2 = 7 and ABCDEF = 225475