Puzzle for March 16, 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
In eq.4, replace F with B + E (from eq.2): E + B + E = B + C which becomes 2×E + B = B + C Subtract B from each side of the equation above: 2×E + B – B = B + C – B which makes 2×E = C
Hint #2
Subtract the left and right sides of eq.3 from the left and right sides of eq.6, respectively: B + C – (B + D) = A + D – (A + E) which becomes B + C – B – D = A + D – A – E which becomes C – D = D – E Add D and E to both sides of the above equation: C – D + D + E = D – E + D + E which becomes eq.6a) C + E = 2×D
Hint #3
In eq.6a, replace C with 2×E: 2×E + E = 2×D which becomes 3×E = 2×D Divide both sides of the above equation by 2: 3×E ÷ 2 = 2×D ÷ 2 which makes 1½×E = D
Hint #4
In eq.5, substitute 2×E for C, 1½×E for D, and B + E for F (from eq.2): 2×E + 1½×E = B + B + E which becomes 3½×E = 2×B + E Subtract E from each side of the above equation: 3½×E – E = 2×B + E – E which makes 2½×E = 2×B Divide both sides by 2: 2½×E ÷ 2 = 2×B ÷ 2 which makes 1¼×E = B
Hint #5
Substitute 1¼×E for B in eq.2: 1¼×E + E = F which makes 2¼×E = F
Hint #6
Substitute 1¼×E for B, and 1½×E for D in eq.3: 1¼×E + 1½×E = A + E which becomes 2¾×E = A + E Subtract E from each side of the equation above: 2¾×E – E = A + E – E which makes 1¾×E = A
Solution
Substitute 1¾×E for A, 1¼×E for B, 2×E for C, 1½×E for D, and 2¼×E for F in eq.1: 1¾×E + 1¼×E + 2×E + 1½×E + E + 2¼×E = 39 which simplifies to 9¾×E = 39 Divide both sides of the above equation by 9¾: 9¾×E ÷ 9¾ = 39 ÷ 9¾ which means E = 4 making A = 1¾×E = 1¾ × 4 = 7 B = 1¼×E = 1¼ × 4 = 5 C = 2×E = 2 × 4 = 8 D = 1½×E = 1½ × 4 = 6 F = 2¼×E = 2¼ × 4 = 9 and ABCDEF = 758649