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The Big and the Small Kingdom are both rectangular islands and divided into rectangular landscape. In each province there is a road that runs along one of the diagonals. On each island exist roads that make a closed route, which does not go through any point several times. The picture shows the Little Kingdom, which has six area:

enter image description here

The Great Kingdom has an odd number of landscapes. How many landscapes does the Great Empire have at least?

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  • $\begingroup$ I'm guessing you're using kingdom and empire as being the same thing? $\endgroup$ – DrunkWolf Dec 13 '15 at 13:28
  • $\begingroup$ Are "landscape" and "province" and "area" the same thing? Are "Big" and "Great" the same thing? $\endgroup$ – Kevin Dec 14 '15 at 17:11
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The answer is

9

One way to do this is:

solution

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  • $\begingroup$ Can you prove that there are no solutions for islands with 1, 3, 5, or 7 provinces? $\endgroup$ – Kevin Dec 15 '15 at 15:07
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The answer is

5, in case it's ok to not go through all squares.

In the general case

We note that you can't possible have a path with less then 4 squares.

In the case where we don't have to go through all squares

5 is the lowest odd number higher then 4, so if we can find a solution, we have proof that this is the optimal solution. Luckily such a solution is easily found
enter image description here

sidenote:

Just because the Big kingdom has less landscapes then the Small kingdom doesn't mean they're smaller.

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  • $\begingroup$ Why is it impossible? $\endgroup$ – algebra1 Dec 13 '15 at 21:04
  • $\begingroup$ Let's not waste time on that. It is possible for odd numbers, the question is pointed at "what is the smallest [...]" which indicates that it does exist a solution. I can post one later if you want. $\endgroup$ – algebra1 Dec 13 '15 at 21:14
  • $\begingroup$ @algebra1 i'd like to see that to be honest, love to be wrong :) That said, i did provide a solution for the question you had. $\endgroup$ – DrunkWolf Dec 13 '15 at 21:17
  • $\begingroup$ Love your solution too, only it is not a correct one. In the text it says that every landscape has a road within it. I'll post a picture of a solution. $\endgroup$ – algebra1 Dec 13 '15 at 21:20
  • $\begingroup$ @algebra1 it is correct as written, sure you say each landscape has a road, but you only ask for a closed road, not one that uses all subroads $\endgroup$ – DrunkWolf Dec 13 '15 at 22:30

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