yes, there is always integer territory, integer captures, and integer stones, meaning itās always an integer under both Territory (Japanese) and Area (Chinese/NZ) scoring without komi

only if there is a resignation

It has to do with a certain CGT idea. But the idea is simple: if there is a stack of coupons of linearly descending values (say 20,19,18,17,16, etc), and weāll assume for a second that this is not accompanied by another game, and assuming there is an *even* number of coupons, the difference between each coupon and the next is a certain value d (d=1 in the example given).

so, first take the simplest stack: a stack of two coupons, one has value 2d, and the next has value d (weāre making the floor equal to d just for simplicity), so if you go first, you take value 2d, and they take value d, resulting in a margin of 2d-d = d, which is half the value of the largest coupon on the stack, weāll see that thatās not just a fluke.

Next up weāll look at four coupons: 4d, 3d, 2d, and d. A takes 4d, B takes 3d, A takes 2d, and B takes d, the A has 4d+2d = 6d, B has 3d+d = 4d, meaning A wins by 6d-4d = 2d once again half the size of the largest coupon.

Now, letās show this in a more general case, where the stack is made of Nd, (N-1)d, (N-2)d, (N-3)d, ā¦, 3d, 2d, d, where N is an even number. Weāre gonna skip the part where we go over who gets what and the total, but instead get the margin immediately through Nd - (N-1)d + (N-2)d - (N-3)d + ā¦ - 3d + 2d - d. We can group these together like [Nd - (N-1)d] + [(N-2)d - (N-3)d]+ā¦+[4d - 3d] + [2d - d], one thing youāll notice is that each group is equal to d, making the resulting amount d+d+ā¦d+d, where there are half as many dās as there were coupons on the stack, resulting in a marginal value of (N/2)d.

so if we have a largest coupon of any stack that goes to 0, weāll call this value V, where V=Nd, the amount A gets for going first is V/2 = (N/2)d.

Thereās a slightly different calculation if the smallest coupon is *not* d (I originally said it didnāt matter, but the truth is that itās about the size of the stack and not the largest coupon in that case)