Puzzle for October 27, 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 C + F with A + E (from eq.2): B + E = A + A + E which becomes B + E = 2×A + E Subtract E from both sides of the above equation: B + E – E = 2×A + E – E which makes B = 2×A
Hint #2
In eq.3, replace B with 2×A: E – A = 2×A – E Add A and E to each side of the equation above: E – A + A + E = 2×A – E + A + E which makes 2×E = 3×A Divide both sides by 2: 2×E ÷ 2 = 3×A ÷ 2 which makes E = 1½×A
Hint #3
In eq.6, substitute 2×A for B, and 1½×A for E: 2×A + D – 1½×A = 1½×A – A – D which becomes ½×A + D = ½×A – D In the above equation, subtract ½×A from both sides, and add D to both sides: ½×A + D – ½×A + D = ½×A – D – ½×A + D which makes 2×D = 0 which means D = 0
Hint #4
Substitute 2×A for B, 0 for D, and 1½×A for E in eq.4: 2×A – C – 0 = C + 0 – 1½×A which becomes 2×A – C = C – 1½×A Add C and 1½×A to both sides of the equation above: 2×A – C + C + 1½×A = C – 1½×A + C + 1½×A which makes 3½×A = 2×C Divide both sides by 2: 3½×A ÷ 2 = 2×C ÷ 2 which makes 1¾×A = C
Hint #5
Substitute 1¾×A for C, and 1½×A for E in eq.2: 1¾×A + F = A + 1½×A which becomes 1¾×A + F = 2½×A Subtract 1¾×A from each side of the above equation: 1¾×A + F – 1¾×A = 2½×A – 1¾×A which makes F = ¾×A
Solution
Substitute 2×A for B, 1¾×A for C, 0 for D, 1½×A for E, and ¾×A for F in eq.1: A + 2×A + 1¾×A + 0 + 1½×A + ¾×A = 28 which simplifies to 7×A = 28 Divide both sides of the above equation by 7: 7×A ÷ 7 = 28 ÷ 7 which means A = 4 making B = 2×A = 2 × 4 = 8 C = 1¾×A = 1¾ × 4 = 7 E = 1½×A = 1½ × 4 = 6 F = ¾×A = ¾ × 4 = 3 and ABCDEF = 487063