Puzzle the neighbor

1. Puzzle | The neighbor
Hup university
17-02-2018

2. Puzzle problem
• Sahil has got a challenge problem from his
friend Akhil. Sahil has to find all values of n for
which one can fill an n × n table with +’s and -
’s (one per cell) so that every cell
has exactly one neighbor with the opposite
sign. Two cells are considered neighbors if
they are either in the same row or the same
column. Can you help out Sahil to find such
values of n ?

3. Solution
• We will scan row by row and find the value of
each cell using three rules 2 neighbors, 3
neighbors , 4 neighbors.

4. Rule for 2 neighbors
A B
C
A/1
B/1 C/0
If A=1 and B=1 and C=0

5. First step
1 1 ? ? ? ?
0 ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
2 neighbors rule
A=1 then B=1,C=0

6. Rule of 3 neighbors
B A D
C
A
B C D
If A=1 then B=1, D=1 C=0
Or A=1 and B=1 then C=0 , D=1

7. Second step
1 1 1 ? ? ?
0 0 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
3 neighbors rule
A,B=1 then D=1, C=0

8. Rule of 4 neighbors
C
B A E
D
A
B C D E
If A=1 then one neighbor will be 0 other will be 1
Or If A=0 then one neighbor will be 1 other will be 0

9. 4 neighbors
1 1 1 ? ? ?
0 0 0 ? ? ?
? 0 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
4 neighbors rule
A=0 , one neighbor will be
1 other will be 0

10. Third step
1 1 1 1 ? ?
0 0 0 ? ? ?
? 0 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
3 neighbors rule

11. 4th step
1 1 1 1 1 ?
0 0 0 0 ? ?
? 0 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
3 neighbors rule

12. 5th step
1 1 1 1 1 1
0 0 0 0 0 ?
? 0 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
3 neighbors rule

13. 6th step
1 1 1 1 1 1
0 0 0 0 0 0
? 0 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
2 neighbors rule

14. 7th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
3neighbors rule

15. 8th step
1 1 1 1 1 1
0 0 0 0 0 ?
0 0 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
4neighbors rule

16. 9th step
1 1 1 1 1 1
0 0 0 0 0 ?
0 0 0 ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
4neighbors rule

17. 10th step
1 1 1 1 1 1
0 0 0 0 0 ?
0 0 0 0 ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
4neighbors rule

18. 11th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
4neighbors rule

19. 12th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
3neighbors rule

20. 13th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
3neighbors rule

21. 14th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
4 neighbors rule

22. 15th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
4 neighbors rule

23. 16th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 ? ?
? ? ? ? ? ?
? ? ? ? ? ?
4neighbors rule

24. 17th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 ?
? ? ? ? ? ?
? ? ? ? ? ?
4neighbors rule

25. 18th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
? ? ? ? ? ?
? ? ? ? ? ?
3neighbors rule

26. 19th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 ? ? ? ? ?
? ? ? ? ? ?
3neighbors rule

27. 20th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 ? ? ? ?
? ? ? ? ? ?
4neighbors rule

28. 21th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 ? ? ?
? ? ? ? ? ?
4neighbors rule

29. 22th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 ? ?
? ? ? ? ? ?
4neighbors rule

30. 23th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 ?
? ? ? ? ? ?
4 neighbors rule

31. 24th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
? ? ? ? ? ?
3neighbors rule

32. 25th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
0 ? ? ? ? ?
3neighbors rule

33. 26th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
0 0 ? ? ? ?
4 neighbors rule

34. 27th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
0 0 0 ? ? ?
4 neighbors rule

35. 28th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
0 0 0 0 ? ?
4neighbors rule

36. 29th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
0 0 0 0 0 ?
4neighbors rule

37. 30th step
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
0 0 0 0 0 0
3neighbors rule

38. End row by row scan
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1 1
0 0 0 0 0 0