# Mad Numbers for Pn x Pn

Let Pn be the path on n vertices, and consider playing Lights Out! on Pn x Pn. Is it always possible to switch all lights off, starting from an arbitrary position? Let N be the set of numbers n+1 such that the answer is no on Pn x Pn. Any multiple of a number in N is again in N. Below we give the numbers in N that are not multiples of other elements in N.

76 elements of N below 10^9 are known, namely

```#1: 5				(2^2+1)
#2: 6
#3: 17				(2^4+1)
#4: 31				(2^5-1)
#5: 33				(2^5+1)
#6: 63				(2^6-1)
#7: 127				(2^7-1)
#8: 129				(2^7+1)
#9: 171				(2^9+1)/3
#10: 257			(2^8+1)
#11: 511			(2^9-1)
#12: 683			(2^11+1)/3
#13: 2047			(2^11-1)
#14: 2731			(2^13+1)/3
#15: 2979			(2^15+1)/11
#16: 3277			(2^14+1)/5
#17: 3641			(2^15+1)/9
#18: 8191			(2^13-1)
#19: 28197			(2^24-1)/595
#20: 43691			(2^17+1)/3
#21: 48771			(2^21+1)/43
#22: 52429			(2^18+1)/5
#23: 61681			(2^20+1)/17
#24: 65537			(2^16+1)
#25: 85489			(2^26+1)/785
#26: 131071			(2^17-1)
#27: 174763			(2^19+1)/3
#28: 178481			(2^23-1)/47
#29: 233017			(2^21+1)/9
#30: 253241			(2^26+1)/265
#31: 256999			(2^29-1)/2089
#32: 486737			(2^29-1)/1103
#33: 524287			(2^19-1)
#34: 704093			(2^30+1)/1525
#35: 784899			(2^27+1)/171
#36: 838861			(2^22+1)/5
#37: 1016801			(2^25+1)/33
#38: 1082401			(2^25-1)/31
#39: 1609167			(2^36-1)/42705
#40: 1657009			(2^27+1)/81
#41: 1838599			(2^27-1)/73
#42: 1965379			(2^36-1)/34965
#43: 2304167			(2^29-1)/233
#44: 2792907			(2^36-1)/24605
#45: 2796203			(2^23+1)/3
#46: 3033169			(2^29+1)/177
#47: 3303821			(2^30+1)/325
#48: 3605429			(2^34+1)/4765
#49: 3705353			(2^35+1)/9273
#50: 6700417			(2^32+1)/641
#51: 7949043			(2^36-1)/8645
#52: 8727391			(2^35-1)/3937
#53: 9335617			(2^36+1)/7361
complete up to n=10^7
#54: 12576771			(2^33+1)/683
#55: 13788017			(2^33-1)/623
#56: 15790321			(2^28+1)/17
#57: 19173961			(2^27-1)/7
#58: 21225581			(2^42+1)/207205
#59: 24214051			(2^35+1)/1419
#60: 25080101			(2^34+1)/685
#61: 25781083			(2^37+1)/5331
#62: 53353631			(2^33-1)/161
#63: 102964687			(2^45-1)/341713
#64: 120296677			(2^38+1)/2285
#65: 164511353			(2^41-1)/13367
#66: 169533009			(2^45+1)/207537
#67: 201302019			(2^39+1)/2731
#68: 207207011			(2^45+1)/169803
#69: 240068041			(2^38+1)/1145
#70: 256957153			(2^45-1)/136927
#71: 464955857			(2^46+1)/151345
#72: 598781009			(2^42+1)/7345
#73: 616318177			(2^37-1)/223
#74: 715827883			(2^31+1)/3
#75: 905040953			(2^43-1)/9719
#76: 993089953			(2^54+1)/18139745
complete up to n=10^9 for pmorder < 200
```

Note that the MAD numbers larger than 3 are precisely the numbers not divisible by 3 on this list.

There is also information on a rectangular board.