Puzzle for May 30, 2023 ( )
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.4, substitute (D + F) for E (from eq.3): C - (D + F) = D - F which becomes C - D - F = D - F Add D and F to both sides of the above equation: C - D - F + D + F = D - F + D + F which makes eq.4a) C = 2×D
Hint #2
In eq.2, replace C with 2×D: B = 2×D + D which makes eq.2a) B = 3×D
Hint #3
In eq.6, replace B with 3×D, and C with 2×D: D + E + F = 3×D + 2×D - E which becomes D + E + F = 5×D - E In the above equation, subtract D from both sides, and add E to both sides: D + E + F - D + E = 5×D - E - D + E which becomes eq.6a) 2×E + F = 4×D
Hint #4
In eq.6a, substitute (D + F) for E (from eq.3): 2×(D + F) + F = 4×D which becomes 2×D + 2×F + F = 4×D which becomes 2×D + 3×F = 4×D Subtract 2×D from each side of the above equation: 2×D + 3×F - 2×D = 4×D - 2×D which makes 3×F = 2×D Divide both sides by 2: 3×F ÷ 2 = 2×D ÷ 2 which makes 1½×F = D
Hint #5
Substitute (1½×F) for D in eq.4a: C = 2×(1½×F) which makes C = 3×F
Hint #6
Substitute (1½×F) for D in eq.2a: B = 3×(1½×F) which makes B = 4½×F
Hint #7
Substitute 1½×F for D in eq.3: E = 1½×F + F which makes E = 2½×F
Hint #8
Substitute 3×F for C, 2½×F for E, and 1½×F for D in eq.5: A - 3×F = 2½×F - 1½×F which becomes A - 3×F = F Add 3×F to both sides of the equation above: A - 3×F + 3×F = F + 3×F which makes A = 4×F
Solution
Substitute 4×F for A, 4½×F for B, 3×F for C, 1½×F for D, and 2½×F for E in eq.1: 4×F + 4½×F + 3×F + 1½×F + 2½×F + F = 33 which simplifies to 16½×F = 33 Divide both sides of the above equation by 16½: 16½×F ÷ 16½ = 33 ÷ 16½ which means F = 2 making A = 4×F = 4 × 2 = 8 B = 4½×F = 4½ × 2 = 9 C = 3×F = 3 × 2 = 6 D = 1½×F = 1½ × 2 = 3 E = 2½×F = 2½ × 2 = 5 and ABCDEF = 896352