#OK to post
#Homework 26, Rebecca Embar

#OK to post
#Homework 26, Rebecca Embar

read `C26.txt`:

#3
SGwyt:=proc(a,b) local C,x,c:
	option remember:
	C := {seq([x,b],x=0..a-1),seq([a,x],x=0..b-1),seq([a-x,b-x],x=1..min(a,b))}:
	mex({seq(SGwyt(op(c)),c in C)}):
end:

#4
FindUP:=proc(L) local n,LCut,L1,p,i,L2:
	for n from 1 to nops(L) do 
		LCut:=[seq(L[i],i=n..nops(L))]:
		L1:=[seq(L[i],i=1..n-1)]:
		for p from 1 to trunc(nops(LCut)/2) do
			L2:=[seq(LCut[i],i=1..p)]:
			if LCut=[seq(op(L2),i=1..trunc(nops(LCut)/p)),seq(L2[i],i=1..(nops(LCut) mod p))] then 
				return(L1,L2):
			fi:
		od:
	od:
	FAIL:
end:

#5
#First 10 terms of seq(FindUP([seq(SGwyt(i,b)-b,b=0..50)])[2],i=0..10)
#i=0 [0] 
#i=1 [1,1,-2]
#i=2 [2,-1,-1]
#i=3 [2,3,-2,-4,3,-2]
#i=4 [-1,3,1,-1,3,1,-5,3,1,-1,-5,1]
#i=5 [-2,2,3,-2,2,3,-2,2,-6,-2,2,3,-2,2,3,-2,2,3,-7,2,3,-2,2,-7]
#i=6 [4,4,4,4,4,-8,-8,4,4,4,-8,-8]
#i=7 [5,-3,-8], 
#i=8 [6,-4,-4,5,5,-3,4,6,-5,5,-3,5,5,5,-9,-11,-4,6,6,6,-5,-10,-3,-3],
#i=9 [-3,-6,-4,6,-5,5,7,-3,6,6,-5,-5,6,-6,-6,-3,4,-6,7,-3,6,6,2,-6]

#6
#FindUP([seq(SGwyt(5,b)-b,b=0..300)]) outputs
#L1=[5, 2, 2, -3, 2, 3, 4, -6, -6, -2, 2, 3, -3, 2, 3, -2, 2, -6, -2, 2, 3, -2, 2, 3, -2, 2, -6]
#L2=[-2, 2, 3, -2, 2, 3, -2, 2, -6, -2, 2, 3, -2, 2, 3, -2, 2, 3, -7, 2, 3, -2, 2, -7]
#So conjecturally, after the first 27 entries for b from 0 to 26,
#SGwyt(5,b)-b follows the pattern given in L2, which has 24 entries 
#(10^100-26) mod 24 = 14, and L2[14]=2
#So SGwyt(5,10^100)=2+10^100