Puzzle for March 4, 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
Add A to both sides of eq.4: A + D + A = B – A + A which becomes eq.4a) 2×A + D = B In eq.5, replace B with 2×A + D (from eq.4a): D + E = A + 2×A + D + C – D which becomes eq.5a) D + E = 3×A + C
Hint #2
In eq.5a, replace E with C + D (from eq.2): D + C + D = 3×A + C which becomes C + 2×D = 3×A + C Subtract C from both sides of the above equation: C + 2×D – C = 3×A + C – C which makes 2×D = 3×A Divide both sides of the equation above by 2: 2×D ÷ 2 = 3×A ÷ 2 which makes D = 1½×A
Hint #3
In eq.4a, replace D with 1½×A: 2×A + 1½×A = B which makes 3½×A = B
Hint #4
In eq.3, substitute 3½×A for B, and 1½×A for D: 3½×A – C = 1½×A – A which becomes 3½×A – C = ½×A In the equation above, add C to each side, and subtract ½×A from each side: 3½×A – C + C – ½×A = ½×A + C – ½×A which makes 3×A = C
Hint #5
Substitute 3×A for C, and 1½×A for D in eq.2: 3×A + 1½×A = E which makes 4½×A = E
Hint #6
Substitute 3½×A for B, 3×A for C, and 1½×A for D in eq.6: 3½×A + 3×A – 1½×A = 1½×A + F which becomes 5×A = 1½×A + F Subtract 1½×A from each side of the above equation: 5×A – 1½×A = 1½×A + F – 1½×A which makes 3½×A = F
Solution
Substitute 3½×A for B and F, 3×A for C, 1½×A for D, and 4½×A for E in eq.1: A + 3½×A + 3×A + 1½×A + 4½×A + 3½×A = 34 which simplifies to 17×A = 34 Divide both sides of the above equation by 17: 17×A ÷ 17 = 34 ÷ 17 which means A = 2 making B = F = 3½×A = 3½ × 2 = 7 C = 3×A = 3 × 2 = 6 D = 1½×A = 1½ × 2 = 3 E = 4½×A = 4½ × 2 = 9 and ABCDEF = 276397