Universal Traps

I use the terms universal trap, ubiquitous trap and UT interchangeably. I am referring to a sphere consisting of two original spots (a type 1 UT) or of two original spots plus a pivot which also communicates with a live spot outside of the sphere (a type 2 UT).

A type 1 UT is a biosphere and is a legitimate trap. In the four spot normal game after 4+ 1(5)1[2] 1(6@2)5, the sphere S(3) is a type 1 universal trap. If this sphere were now to be played as a separate game, the second player of that game could force his choice of an even or an odd number of survivors and thus would win in both normal and misere sprouts. The entire position therefore comes down to the sphere S(2). We can think of the sphere S(2) as a separate, normal game, and whoever loses this separate game will be forced to be the “first player” in the S(3) game and thus will lose the entire position. Since the sphere S(2) has two cannibals and will produce one survivor, Left will win the S(2) game and thus the entire position.

A type 2 UT is not a biosphere and is therefore not a legitimate trap, but this formation behaves very much like a trap. Consider the normal eight spot game after 8+ 1(9)1[2] 1(10)9[3,4]. The sphere S(3) is a type 2 universal trap. This sphere is not a biosphere because spot 10 communicates with live spots that belong to a different sphere. But the sphere S(3) behaves very much like a trap. Any move exclusively to S(3) converts S(3) into a switch. The main functional difference between a type 2 UT and a proper trap is the possibility, which always must be considered, that the move from pivot to internal spot might convert the exterior sphere into a switch also, creating double switches. In the present position, for instance, we must ask ourselves whether the move 3(11)10 converts the sphere S(5) into a switch. The answer is no. The sphere S(5) produces three survivors whether or not spot 10 plays into the sphere. So in the present case the type 2 universal trap is quite nearly equivalent to a legitimate trap. After 8+ 1(9)1[2] 1(10)9[3,4], Left is lost. The sphere S(2,5) has six cannibals and will produce four survivors. Left will be forced to move to the universal trap, whereupon Right will cause that sphere to produce two survivors.

There is a crucial subtlety in the definition of the type 2 UT. It is mandatory that the pivot communicate with a live spot beyond the sphere. For example, in the normal four spot game after 4+ 1(5)1[2] 1(6@3)5, the sphere S(3) is not a universal trap at all. The pivot, spot 6, does not have access to any spot outside of S(3). S(3) is not a UT and is not a trap. S(3) simply produces two survivors.

Back to World Game of Sprouts Association Home Page