Puzzle for September 7, 2022 ( )
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
eq.6 may be written as: A + B = (D + E) ÷ 2 Multiply both sides of the above equation by 2: 2 × (A + B) = 2 × (D + E) ÷ 2 which becomes eq.6a) 2×A + 2×B = D + E Add A and B to both sides of eq.3: E - A + A + B = A - B + A + B which becomes eq.3a) E + B = 2×A
Hint #2
In eq.6a, replace 2×A with E + B (from eq.3a): E + B + 2×B = D + E which becomes E + 3×B = D + E Subtract E from each side of the above equation: E + 3×B - E = D + E - E which makes 3×B = D
Hint #3
Add F and E to both sides of eq.4: C - F + F + E = D - E + F + E which becomes C + E = D + F Subtract the left and right sides of the equation above from the left and right sides of eq.2: B + C - (C + E) = E + F - (D + F) which becomes B + C - C - E = E + F - D - F which becomes eq.2a) B - E = E - D
Hint #4
In eq.2a, substitute 3×B for D: B - E = E - 3×B Add E and 3×B to both sides of the above equation: B - E + E + 3×B = E - 3×B + E + 3×B which makes 4×B = 2×E Divide both sides by 2: 4×B ÷ 2 = 2×E ÷ 2 which makes 2×B = E
Hint #5
Substitute 2×B for E in eq.3a: 2×B + B = 2×A which makes 3×B = 2×A Divide both sides of the above equation by 2: 3×B ÷ 2 = 2×A ÷ 2 which makes 1½×B = A
Hint #6
Substitute 2×B for E in eq.5: F - 2×B = 2×B - B which becomes F - 2×B = B Add 2×B to both sides of the equation above: F - 2×B + 2×B = B + 2×B which makes F = 3×B
Hint #7
Substitute 3×B for F and D, and 2×B for E in eq.4: C - 3×B = 3×B - 2×B which becomes C - 3×B = B Add 3×B to both sides of the above equation: C - 3×B + 3×B = B + 3×B which makes C = 4×B
Solution
Substitute 1½×B for A, 4×B for C, 3×B for D and F, and 2×B for E in eq.1: 1½×B + B + 4×B + 3×B + 2×B + 3×B = 29 which simplifies to 14½×B = 29 Divide both sides of the above equation by 14½: 14½×B ÷ 14½ = 29 ÷ 14½ which means B = 2 making A = 1½×B = 1½ × 2 = 3 C = 4×B = 4 × 2 = 8 D = F = 3×B = 3 × 2 = 6 E = 2×B = 2 × 2 = 4 and ABCDEF = 328646