superprismatic

Well, it seems nobody wants to provide a proof so

I'm not using spoilers because this problem is not a puzzle with a known answer. I checked your answer and for your V=(1,1,1,1,-1,-1,-1,-1), I also compute Q(V) to be 6898. Notice that, because of the absolute value in the function Q, Q(V)=Q(-V). So, Q(V)=6898 for V=(-1,-1,-1,-1,1,1,1,1) as well. Will you give a short description of what your program does? Is there a lot of guessing? Is a genetic algorithm involved? Did you use one of the methods (like the Simplex Algorithm) to optimize Q(V)? Also, do you have any clue as to why the V you got is so pretty? Perhaps it's a property of Pi itself. I hope we can get even better answers than the pretty one you found.

Getting zero is probably an indication of a bug. After all, Pi is pretty random!

Let me know if this shouldn't be happening, and I'll see if I can find out where I've made a mistake in my programming.

What I meant by "all possible positions inside P" was: abs(P(1)*V(1)+P(2)*V(2)+P(3)*V(3)+P(4)*V(4)+P(5)*V(5)+P(6)*V(6)+P(7)*V(7)+P(8)*V(8)+P(9)*V(9)+P(10)*V(10))+ abs(P(2)*V(1)+P(3)*V(2)+P(4)*V(3)+P(5)*V(4)+P(6)*V(6)+P(7)*V(6)+P(8)*V(7)+P(9)*V(8)+P(10)*V(9)+P(11)*V(10))+ abs(P(3)*V(1)+P(4)*V(2)+P(5)*V(3)+P(6)*V(4)+P(7)*V(7)+P(8)*V(6)+P(9)*V(7)+P(10)*V(8)+P(11)*V(9)+P(12)*V(10))+ abs(P(4)*V(1)+P(5)*V(2)+P(6)*V(3)+P(7)*V(4)+P(8)*V(8)+P(9)*V(6)+P(10)*V(7)+P(11)*V(8)+P(12)*V(9)+P(13)*V(10))+ . . . abs(P(992)*V(1)+P(993)*V(2)+P(994)*V(3)+P(995)*V(4)+P(996)*V(8)+P(997)*V(6)+P(998)*V(7)+P(999)*V(8)+P(1000)*V(9)+P(1001)*V(10)) [/code] I was thinking of "sliding" the 10-long V vector thru all consecutive 10-long segments of P. By the way, V can have positive and negative elements but may not have a zero element. Although I have a favorite vector V together with a relatively small Q(V), it may not be the smallest possible. I'm curious to see if anyone can find something better.

Consider the first 1001 digits of pi (ignore the decimal point): 3. 14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679 82148086513282306647093844609550582231725359408128 48111745028410270193852110555964462294895493038196 44288109756659334461284756482337867831652712019091 45648566923460348610454326648213393607260249141273 72458700660631558817488152092096282925409171536436 78925903600113305305488204665213841469519415116094 33057270365759591953092186117381932611793105118548 07446237996274956735188575272489122793818301194912 98336733624406566430860213949463952247371907021798 60943702770539217176293176752384674818467669405132 00056812714526356082778577134275778960917363717872 14684409012249534301465495853710507922796892589235 42019956112129021960864034418159813629774771309960 51870721134999999837297804995105973173281609631859 50244594553469083026425223082533446850352619311881 71010003137838752886587533208381420617177669147303 59825349042875546873115956286388235378759375195778 18577805321712268066130019278766111959092164201989 Now, let P(i) be the ith digit of pi, where i is in [1,1001]. Your task is to find a 10-long vector of non-zero integers V such that the sum of the absolute values of the dot products of V with P at all possible positions inside P produces a minimum value. That is, let Q(V)=sum{i=0,991}[abs(sum{j=1,10}[P(i+j)*V(j)])] where {i=a,b} means 'i varies over all integer values in the closed interval [a,b].' Find V and its Q(V) such that Q(V) is as small as possible.

Good puzzle superprismatic -- did you make this one up? Yes, I made it up. Thanks for telling me you liked it.

Very nice, concise explanation!

By 'mirrored symmetry' I mean a flip is required, e.g., a flip is required to transform this ABCDE FGHIJ KLMNO PQRST UVWXY [/code] into this [code] EDCBA JIHGF ONMLK TSRQP YXWVU as it can't be done using rotations alone. I don't understand your terminology. So, I can't really follow your argument. What are 'rotational segments' and 'full-board rotations'? And I can't see where the 'mirrored' part comes in.