Puzzle for April 13, 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 B with C + D (from eq.3): C + D – D = F – C which becomes C = F – C Add C to both sides of the equation above: C + C = F – C + C which makes eq.5a) 2×C = F
Hint #2
In eq.6, replace F with 2×C: 2×C – E = C + E In the above equation, add E to both sides, and subtract C from both sides: 2×C – E + E – C = C + E + E – C which makes C = 2×E
Hint #3
In eq.5a, substitute (2×E) for C: 2×(2×E) = F which makes 4×E = F
Hint #4
Substitute 4×E for F, D + E for A (from eq.2), and 2×E for C in eq.4: 4×E – D = D + E – 2×E which becomes 4×E – D = D – E Add D and E to both sides of the equation above: 4×E – D + D + E = D – E + D + E which becomes 5×E = 2×D Divide both sides by 2: 5×E ÷ 2 = 2×D ÷ 2 which makes 2½×E = D
Hint #5
Substitute 2½×E for D in eq.2: 2½×E + E = A which makes 3½×E = A
Hint #6
Substitute 2×E for C, and 2½×E for D in eq.3: 2×E + 2½×E = B which makes 4½×E = B
Solution
Substitute 3½×E for A, 4½×E for B, 2×E for C, 2½×E for D, and 4×E for F in eq.1: 3½×E + 4½×E + 2×E + 2½×E + E + 4×E = 35 which simplifies to 17½×E = 35 Divide both sides of the equation above by 17½: 17½×E ÷ 17½ = 35 ÷ 17½ which means E = 2 making A = 3½×E = 3½ × 2 = 7 B = 4½×E = 4½ × 2 = 9 C = 2×E = 2 × 2 = 4 D = 2½×E = 2½ × 2 = 5 F = 4×E = 4 × 2 = 8 and ABCDEF = 794528