Puzzle for May 6, 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.3, add A to both sides, and subtract F from both sides: B – A + A – F = C + F + A – F which becomes eq.3a) B – F = C + A Subtract C from both sides of eq.6: B + C – F – C = A + E – C which becomes eq.6a) B – F = A + E – C
Hint #2
In eq.6a, replace B – F with C + A (from eq.3a): C + A = A + E – C In the equation above, subtract A from each side, and add C to each side: C + A – A + C = A + E – C – A + C which makes 2×C = E
Hint #3
Add A to both sides of eq.3: B – A + A = C + F + A which becomes B = C + F + A which may be written B = C + A + F In the above equation, replace A + F with C (from eq.4): B = C + C which makes B = 2×C and also means eq.3b) E = B = 2×C
Hint #4
In eq.2, substitute B for E: B + F = A + B Subtract B from each side of the equation above: B + F – B = A + B – B which makes F = A
Hint #5
Substitute A for F in eq.4: C = A + A which makes C = 2×A
Hint #6
Substitute 2×A for C in eq.3b: E = B = 2×(2×A) which makes E = B = 4×A
Hint #7
Substitute 2×A for C, and 4×A for E in eq.5: D = A + 2×A – D + 4×A which becomes D = 7×A – D Add D to each side of the equation above: D + D = 7×A – D + D which becomes 2×D = 7×A Divide both sides by 2: 2×D ÷ 2 = 7×A ÷ 2 which makes D = 3½×A
Solution
Substitute 4×A for B and E, 2×A for C, 3½×A for D, and A for F in eq.1: A + 4×A + 2×A + 3½×A + 4×A + A = 31 which simplifies to 15½×A = 31 Divide both sides of the equation above by 15½: 15½×A ÷ 15½ = 31 ÷ 15½ which means A = 2 making B = E = 4×A = 4 × 2 = 8 C = 2×A = 2 × 2 = 4 D = 3½×A = 3½ × 2 = 7 F = A = 2 and ABCDEF = 284782