Puzzle for August 31, 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.3, replace C with B + D (from eq.2): B + D – D = D – B which becomes B = D – B Add B to each side of the equation above: B + B = D – B + B which makes 2×B = D
Hint #2
In eq.2, replace D with 2×B: B + 2×B = C which makes 3×B = C
Hint #3
In eq.4, substitute 2×B for D, and 3×B for C: 2×B + E – B = B + 3×B which becomes B + E = 4×B Subtract B from each side of the equation above: B + E – B = 4×B – B which makes E = 3×B
Hint #4
Substitute 3×B for C and E in eq.6: B + F = 3×B + 3×B – F which becomes B + F = 6×B – F In the above equation, subtract B from both sides, and add F to both sides: B + F – B + F = 6×B – F – B + F which makes 2×F = 5×B Divide both sides by 2: 2×F ÷ 2 = 5×B ÷ 2 which makes F = 2½×B
Hint #5
Substitute 2×B for D, and 2½×B for F in eq.5: A = 2×B + 2½×B which makes A = 4½×B
Solution
Substitute 4½×B for A, 3×B for C and E, 2×B for D, and 2½×B for F in eq.1: 4½×B + B + 3×B + 2×B + 3×B + 2½×B = 32 which simplifies to 16×B = 32 Divide both sides of the above equation by 16: 16×B ÷ 16 = 32 ÷ 16 which makes B = 3 and makes A = 4½×B = 4½ × 2 = 9 C = E = 3×B = 3 × 2 = 6 D = 2×B = 2 × 2 = 4 F = 2½×B = 2½ × 2 = 5 and ABCDEF = 926465