figured it out
Aug. 1st, 2001 12:01 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
hugmelast night I was up thinking about an interesting problem. I am sure you have all seen this puzzle:
where you have to draw this shape without crossing over any lines or without picking up your pencil. well last night I figured out a formula for doing this. I got up and had grrl hand me my paper and pencil and drew it out... the formula is in several parts.. here are some of the solutions:
the square: L=left, R=right, T=top, B=bottom
the X: V=top, /\= bottom, >=left, <=right
the steeple: ^=top
1. L . V . T . ^ . R . /\ . B
2. B . /\ . R . ^ . T . V . L
3. R . V . T . ^ . L . /\ . B
4. B . /\ . L . ^ . T . V . R
5. L . V . T . ^ . R . B . /\
6. /\ . B . R . ^ . T . V . L
7. R . V . T . ^ . L . B . /\
8. /\ . B . L . ^ . T . V . R
9. L . ^ . T . V . R . /\ . B
10. B . /\ . R . V . T . ^ . L
11. R . ^ . T . V . L . /\ . B
12. B . /\ . L . V . T . ^ . R
13. L . ^ . T . V . R . B . /\
14. /\ . B . R . V . T . ^ . L
15. R . ^ . T . V . L . B . /\
16. /\ . B . L . V . T . ^ . R
17. B . > . T . < . R . ^ . L
18. L . ^ . R . < . T . > . B
19. B . > . T . < . L . ^ . R
20. R . ^ . L . < . T . > . B
21. B . L . ^ . R . < . T . >
22. > . T . < . R . ^ . L . B
23. B . R . ^ . L . < . T . >
24. > . T . < . L . ^ . R . B
25. L . ^ . R . < . T . > . B
26. B . > . T . < . R . ^ . L
27. R . ^ . L . < . T . > . B
28. B . > . T . < . L . ^ . R
29. L . ^ . R . B . > . T . <
30. < . T . > . B . R . ^ . L
29. R . ^ . L . B . > . T . <
30. < . T . > . B . L . ^ . R
This is all I felt like scribbling down. there are many others. Parts of the formula I have made obvious. every solution you discover has 3 neighbors to it. because once you find a solution you can always do it backwards. and there is always a mirror of it (switch the L and the R) that is why you see groups of 4.
because I have written down the solutions this is going to be much more obvious.. but notice there are 2 things you never see in the first or last collum, a T and a ^. why is that.. if you look at the bottom of the box there is a point in which 3 lines come together. between your first and second move you are going to 'create' an end point, this HAS to be on one of the 3 point sides. pretty simple.
now for your 4 point turns always go in with an even number of ways to get out. only go in if there are 1 or 3 ways out. because there are only 2 places possible to create an end point you can't create it at your 4 point turn.
the rest of the formula I am not really sure how to write down, I will have to think about it for a while... I just kind of see it as a shape right now that I can't explain.
happy solving.
where you have to draw this shape without crossing over any lines or without picking up your pencil. well last night I figured out a formula for doing this. I got up and had grrl hand me my paper and pencil and drew it out... the formula is in several parts.. here are some of the solutions:
the square: L=left, R=right, T=top, B=bottom
the X: V=top, /\= bottom, >=left, <=right
the steeple: ^=top
1. L . V . T . ^ . R . /\ . B
2. B . /\ . R . ^ . T . V . L
3. R . V . T . ^ . L . /\ . B
4. B . /\ . L . ^ . T . V . R
5. L . V . T . ^ . R . B . /\
6. /\ . B . R . ^ . T . V . L
7. R . V . T . ^ . L . B . /\
8. /\ . B . L . ^ . T . V . R
9. L . ^ . T . V . R . /\ . B
10. B . /\ . R . V . T . ^ . L
11. R . ^ . T . V . L . /\ . B
12. B . /\ . L . V . T . ^ . R
13. L . ^ . T . V . R . B . /\
14. /\ . B . R . V . T . ^ . L
15. R . ^ . T . V . L . B . /\
16. /\ . B . L . V . T . ^ . R
17. B . > . T . < . R . ^ . L
18. L . ^ . R . < . T . > . B
19. B . > . T . < . L . ^ . R
20. R . ^ . L . < . T . > . B
21. B . L . ^ . R . < . T . >
22. > . T . < . R . ^ . L . B
23. B . R . ^ . L . < . T . >
24. > . T . < . L . ^ . R . B
25. L . ^ . R . < . T . > . B
26. B . > . T . < . R . ^ . L
27. R . ^ . L . < . T . > . B
28. B . > . T . < . L . ^ . R
29. L . ^ . R . B . > . T . <
30. < . T . > . B . R . ^ . L
29. R . ^ . L . B . > . T . <
30. < . T . > . B . L . ^ . R
This is all I felt like scribbling down. there are many others. Parts of the formula I have made obvious. every solution you discover has 3 neighbors to it. because once you find a solution you can always do it backwards. and there is always a mirror of it (switch the L and the R) that is why you see groups of 4.
because I have written down the solutions this is going to be much more obvious.. but notice there are 2 things you never see in the first or last collum, a T and a ^. why is that.. if you look at the bottom of the box there is a point in which 3 lines come together. between your first and second move you are going to 'create' an end point, this HAS to be on one of the 3 point sides. pretty simple.
now for your 4 point turns always go in with an even number of ways to get out. only go in if there are 1 or 3 ways out. because there are only 2 places possible to create an end point you can't create it at your 4 point turn.
the rest of the formula I am not really sure how to write down, I will have to think about it for a while... I just kind of see it as a shape right now that I can't explain.
happy solving.
