101 town problem with connected road
$begingroup$
There are $101$ towns
There are $50$ roads entering each town and $50$ roads leaving each town. Each town is connected with every other town by a one way road. Prove that you can reach one from other by driving along at most two roads.
Please help without graph theoretic solution. I am thinking of applying contradiction. I am not being able to think of a solution.
combinatorics discrete-mathematics graph-theory
edited Dec 20 '18 at 17:06
Maria Mazur
49.9k1361125
asked Jan 12 '18 at 17:24
nandita mukherjeenandita mukherjee
898
$endgroup$
add a comment |
$begingroup$
There are $101$ towns
There are $50$ roads entering each town and $50$ roads leaving each town. Each town is connected with every other town by a one way road. Prove that you can reach one from other by driving along at most two roads.
Please help without graph theoretic solution. I am thinking of applying contradiction. I am not being able to think of a solution.
combinatorics discrete-mathematics graph-theory
edited Dec 20 '18 at 17:06
Maria Mazur
49.9k1361125
asked Jan 12 '18 at 17:24
nandita mukherjeenandita mukherjee
898
$endgroup$
$begingroup$
Maybe try considering the general problem There are $2n+1$ towns and $n$ roads entering each town and $n$ roads leaving each town. Prove that you can reach one from other by driving along at most two roads. Now maybe induction or trying smaller $n$'s first to conclude something (though I'm just throwing ideas).
$endgroup$
– kingW3
Jan 12 '18 at 17:59
add a comment |
$begingroup$
There are $101$ towns
There are $50$ roads entering each town and $50$ roads leaving each town. Each town is connected with every other town by a one way road. Prove that you can reach one from other by driving along at most two roads.
Please help without graph theoretic solution. I am thinking of applying contradiction. I am not being able to think of a solution.
combinatorics discrete-mathematics graph-theory
edited Dec 20 '18 at 17:06
Maria Mazur
49.9k1361125
asked Jan 12 '18 at 17:24
nandita mukherjeenandita mukherjee
898
$endgroup$
There are $101$ towns
There are $50$ roads entering each town and $50$ roads leaving each town. Each town is connected with every other town by a one way road. Prove that you can reach one from other by driving along at most two roads.
Please help without graph theoretic solution. I am thinking of applying contradiction. I am not being able to think of a solution.
combinatorics discrete-mathematics graph-theory
combinatorics discrete-mathematics graph-theory
edited Dec 20 '18 at 17:06
Maria Mazur
49.9k1361125
asked Jan 12 '18 at 17:24
nandita mukherjeenandita mukherjee
898
edited Dec 20 '18 at 17:06
Maria Mazur
49.9k1361125
asked Jan 12 '18 at 17:24
nandita mukherjeenandita mukherjee
898
edited Dec 20 '18 at 17:06
Maria Mazur
49.9k1361125
edited Dec 20 '18 at 17:06
Maria Mazur
49.9k1361125
edited Dec 20 '18 at 17:06
Maria Mazur
49.9k1361125
49.9k1361125
asked Jan 12 '18 at 17:24
nandita mukherjeenandita mukherjee
898
asked Jan 12 '18 at 17:24
nandita mukherjeenandita mukherjee
898
asked Jan 12 '18 at 17:24
nandita mukherjeenandita mukherjee
898
898
$begingroup$
Maybe try considering the general problem There are $2n+1$ towns and $n$ roads entering each town and $n$ roads leaving each town. Prove that you can reach one from other by driving along at most two roads. Now maybe induction or trying smaller $n$'s first to conclude something (though I'm just throwing ideas).
$endgroup$
– kingW3
Jan 12 '18 at 17:59
add a comment |
$begingroup$
Maybe try considering the general problem There are $2n+1$ towns and $n$ roads entering each town and $n$ roads leaving each town. Prove that you can reach one from other by driving along at most two roads. Now maybe induction or trying smaller $n$'s first to conclude something (though I'm just throwing ideas).
$endgroup$
– kingW3
Jan 12 '18 at 17:59
$begingroup$
Maybe try considering the general problem There are $2n+1$ towns and $n$ roads entering each town and $n$ roads leaving each town. Prove that you can reach one from other by driving along at most two roads. Now maybe induction or trying smaller $n$'s first to conclude something (though I'm just throwing ideas).
$endgroup$
– kingW3
Jan 12 '18 at 17:59
$begingroup$
Maybe try considering the general problem There are $2n+1$ towns and $n$ roads entering each town and $n$ roads leaving each town. Prove that you can reach one from other by driving along at most two roads. Now maybe induction or trying smaller $n$'s first to conclude something (though I'm just throwing ideas).
$endgroup$
– kingW3
Jan 12 '18 at 17:59
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Suppose we want to go from $A$ to $B$. If there's a road connected $A$ to $B$ directly, we are done.
Suppose not. Then, there must exist a road from $B$ to $A$.
Now, let $C_1,C_2,...,C_{50}$ be $50$ cities such that there's a road from $A$ to $C_i$.
If there exists $iin { 1,2,...,50}$ such that there's a road from $C_i$ to $B$, we are done.
Suppose for a contradiction there isn't such road. Then for all $1 le i le 50$, each road is from $B$ to $C_i$. But there is also a road from $B$ to $A$ so there are $51$ roads out of $B$, which is a contradiction as required.
edited Jan 12 '18 at 18:43
ArsenBerk
7,61731338
answered Jan 12 '18 at 18:30
nandita mukherjeenandita mukherjee
898
$endgroup$
$begingroup$
This perfectly makes sense! I just edited your answer to clarify what you were saying, I think you can accept your own answer. Good job :)
$endgroup$
– ArsenBerk
Jan 12 '18 at 18:44
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Suppose we want to go from $A$ to $B$. If there's a road connected $A$ to $B$ directly, we are done.
Suppose not. Then, there must exist a road from $B$ to $A$.
Now, let $C_1,C_2,...,C_{50}$ be $50$ cities such that there's a road from $A$ to $C_i$.
If there exists $iin { 1,2,...,50}$ such that there's a road from $C_i$ to $B$, we are done.
Suppose for a contradiction there isn't such road. Then for all $1 le i le 50$, each road is from $B$ to $C_i$. But there is also a road from $B$ to $A$ so there are $51$ roads out of $B$, which is a contradiction as required.
edited Jan 12 '18 at 18:43
ArsenBerk
7,61731338
answered Jan 12 '18 at 18:30
nandita mukherjeenandita mukherjee
898
$endgroup$
$begingroup$
This perfectly makes sense! I just edited your answer to clarify what you were saying, I think you can accept your own answer. Good job :)
$endgroup$
– ArsenBerk
Jan 12 '18 at 18:44
add a comment |
$begingroup$
Suppose we want to go from $A$ to $B$. If there's a road connected $A$ to $B$ directly, we are done.
Suppose not. Then, there must exist a road from $B$ to $A$.
Now, let $C_1,C_2,...,C_{50}$ be $50$ cities such that there's a road from $A$ to $C_i$.
If there exists $iin { 1,2,...,50}$ such that there's a road from $C_i$ to $B$, we are done.
Suppose for a contradiction there isn't such road. Then for all $1 le i le 50$, each road is from $B$ to $C_i$. But there is also a road from $B$ to $A$ so there are $51$ roads out of $B$, which is a contradiction as required.
edited Jan 12 '18 at 18:43
ArsenBerk
7,61731338
answered Jan 12 '18 at 18:30
nandita mukherjeenandita mukherjee
898
$endgroup$
$begingroup$
This perfectly makes sense! I just edited your answer to clarify what you were saying, I think you can accept your own answer. Good job :)
$endgroup$
– ArsenBerk
Jan 12 '18 at 18:44
add a comment |
$begingroup$
Suppose we want to go from $A$ to $B$. If there's a road connected $A$ to $B$ directly, we are done.
Suppose not. Then, there must exist a road from $B$ to $A$.
Now, let $C_1,C_2,...,C_{50}$ be $50$ cities such that there's a road from $A$ to $C_i$.
If there exists $iin { 1,2,...,50}$ such that there's a road from $C_i$ to $B$, we are done.
Suppose for a contradiction there isn't such road. Then for all $1 le i le 50$, each road is from $B$ to $C_i$. But there is also a road from $B$ to $A$ so there are $51$ roads out of $B$, which is a contradiction as required.
edited Jan 12 '18 at 18:43
ArsenBerk
7,61731338
answered Jan 12 '18 at 18:30
nandita mukherjeenandita mukherjee
898
$endgroup$
Suppose we want to go from $A$ to $B$. If there's a road connected $A$ to $B$ directly, we are done.
Suppose not. Then, there must exist a road from $B$ to $A$.
Now, let $C_1,C_2,...,C_{50}$ be $50$ cities such that there's a road from $A$ to $C_i$.
If there exists $iin { 1,2,...,50}$ such that there's a road from $C_i$ to $B$, we are done.
Suppose for a contradiction there isn't such road. Then for all $1 le i le 50$, each road is from $B$ to $C_i$. But there is also a road from $B$ to $A$ so there are $51$ roads out of $B$, which is a contradiction as required.
edited Jan 12 '18 at 18:43
ArsenBerk
7,61731338
answered Jan 12 '18 at 18:30
nandita mukherjeenandita mukherjee
898
edited Jan 12 '18 at 18:43
ArsenBerk
7,61731338
edited Jan 12 '18 at 18:43
ArsenBerk
7,61731338
edited Jan 12 '18 at 18:43
ArsenBerk
7,61731338
7,61731338
answered Jan 12 '18 at 18:30
nandita mukherjeenandita mukherjee
898
answered Jan 12 '18 at 18:30
nandita mukherjeenandita mukherjee
898
answered Jan 12 '18 at 18:30
nandita mukherjeenandita mukherjee
898
898
$begingroup$
This perfectly makes sense! I just edited your answer to clarify what you were saying, I think you can accept your own answer. Good job :)
$endgroup$
– ArsenBerk
Jan 12 '18 at 18:44
add a comment |
$begingroup$
This perfectly makes sense! I just edited your answer to clarify what you were saying, I think you can accept your own answer. Good job :)
$endgroup$
– ArsenBerk
Jan 12 '18 at 18:44
$begingroup$
This perfectly makes sense! I just edited your answer to clarify what you were saying, I think you can accept your own answer. Good job :)
$endgroup$
– ArsenBerk
Jan 12 '18 at 18:44
$begingroup$
This perfectly makes sense! I just edited your answer to clarify what you were saying, I think you can accept your own answer. Good job :)
$endgroup$
– ArsenBerk
Jan 12 '18 at 18:44
add a comment |
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$begingroup$
Maybe try considering the general problem There are $2n+1$ towns and $n$ roads entering each town and $n$ roads leaving each town. Prove that you can reach one from other by driving along at most two roads. Now maybe induction or trying smaller $n$'s first to conclude something (though I'm just throwing ideas).
$endgroup$
– kingW3
Jan 12 '18 at 17:59