- #1
murshid_islam
- 457
- 19
I was reading Ramanujan’s biography (by Kanigel) and there was a mathematical problem in that book which was published in the Strand magazine during the First World War. This is the problem:
"I was talking the other day," said William Rogers to the other villagers gathered around the inn fire, "to a gentleman about the place called Louvain, what the Germans have burnt down. He said he knowed it well – used to visit a Belgian friend there. He said the house of his friend was in a long street, numbered on this side one, two, three, and so on, and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him. Funny thing that! He said he knew there was more than fifty houses on that side of the street, but not so many as five hundred. I made mention of the matter to our parson, and he took a pencil and worked out the number of the house where the Belgian lived I don’t know how he done it."
Perhaps the reader may like to discover the number of that house.
Here is what I tried:
Let n = the number of the house that the Belgian lived in
m = the total number of houses in that street
and 50<m<500
now
[tex]1+2+ \cdots +(n-1) = (n+1)+(n+2)+ \cdots +m[/tex]
[tex]\frac{n(n-1)}{2} = \frac{(n+1+m)(m-n)}{2}[/tex]
[tex]n(n-1) = (n+1+m)(m-n)[/tex]
[tex]2n^2 = m^2 + m[/tex]
now how do i sove for n and m?
"I was talking the other day," said William Rogers to the other villagers gathered around the inn fire, "to a gentleman about the place called Louvain, what the Germans have burnt down. He said he knowed it well – used to visit a Belgian friend there. He said the house of his friend was in a long street, numbered on this side one, two, three, and so on, and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him. Funny thing that! He said he knew there was more than fifty houses on that side of the street, but not so many as five hundred. I made mention of the matter to our parson, and he took a pencil and worked out the number of the house where the Belgian lived I don’t know how he done it."
Perhaps the reader may like to discover the number of that house.
Here is what I tried:
Let n = the number of the house that the Belgian lived in
m = the total number of houses in that street
and 50<m<500
now
[tex]1+2+ \cdots +(n-1) = (n+1)+(n+2)+ \cdots +m[/tex]
[tex]\frac{n(n-1)}{2} = \frac{(n+1+m)(m-n)}{2}[/tex]
[tex]n(n-1) = (n+1+m)(m-n)[/tex]
[tex]2n^2 = m^2 + m[/tex]
now how do i sove for n and m?
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