Puzzle for October 14, 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.
* AB is a 2-digit number (not A×B).
Scratchpad
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Hint #1
In eq.3, subtract C from both sides, and add A to both sides: C + F – C + A = B – A – C + A which becomes F + A = B – C In eq.5, replace B – C with F + A: F + A – D + E = A + F Subtract A and F from both sides of the above equation: F + A – D + E – A – F = A + F – A – F which becomes –D + E = 0 Add D to both sides: –D + E + D = 0 + D which makes E = D
Hint #2
In eq.4, replace E with D: A – C = D – D which becomes A – C = 0 Add C to each side of the above equation: A – C + C = 0 + C which makes A = C
Hint #3
eq.6 may be written as: 10×A + B = B + C + D + E In the above equation, substitute A for C, and D for E: 10×A + B = B + A + D + D which becomes 10×A + B = B + A + 2×D Subtract A and B from both sides: 10×A + B – A – B = B + A + 2×D – A – B which becomes 9×A = 2×D Divide both sides by 2: 9×A ÷ 2 = 2×D ÷ 2 which makes 4½×A = D and also makes E = D = 4½×A
Hint #4
Substitute 4½×A for D, and A for C in eq.2: 4½×A = A + B + A which becomes 4½×A = 2×A + B Subtract 2×A from each side of the equation above: 4½×A – 2×A = 2×A + B – 2×A which makes 2½×A = B
Hint #5
Substitute A for C, and 2½×A for B in eq.3: A + F = 2½×A – A which becomes A + F = 1½×A Subtract A from each side of the above equation: A + F – A = 1½×A – A which makes F = ½×A
Solution
Substitute 2½×A for B, A for C, 4½×A for D and E, and ½×A for F in eq.1: A + 2½×A + A + 4½×A + 4½×A + ½×A = 28 which simplifies to 14×A = 28 Divide both sides of the equation above by 14: 14×A ÷ 14 = 28 ÷ 14 which means A = 2 making B = 2½×A = 2½ × 2 = 5 C = A = 2 D = E = 4½×A = 4½ × 2 = 9 F = ½×A = ½ × 2 = 1 and ABCDEF = 252991