[See below, maybe it should be Contragrasshopper 5.0 and Grasshopper 2.5] M&B06 text and Friedlander applet for Grasshopper Chess make clear that Grasshopper moves over adjacency of course. Thanks for correction. Actually Grasshopper leads off Falcon patent text written 16 yrs ago, Falcon. [Knappen notes Gazelle there too recently as synonym for Camel in a Turkish Great Chess -- not a Hopper.] Of this hopper class, Long Leaper also leaps to capture over adjacent opponent, somewhat unintuitively. Important: not to omit from Hoppers is description the same M&B06 Duniho-named Arrow, which is the same as Dawson-named and invented year 1912 Vao, which is the same as Fourriere-named Canon. I prefer and will continue using ''Canon'' for the hopper piece-type used in Jacks & Witches. This Arrow or Vao or Canon, all the same meaning, is the diagonal equivalent of Cannon. It apparently takes 900 years to get from Cannon, problemist's Pao, to Canon. 100 years ago 1912 saw invention of several other Dawson fairy pieces with Grasshopper and Canon/Vao, both hoppers. ______________________________________________________ As for values, Grasshopper versus Contragrasshopper, one approach is to use informal quick trade-off method by putting them in a C.D.A. imaginatively. Which would you rather have, a row of 8 Gs or 8 CGs? In back rank CG excels and in front rank G excels. Instead of that, the reason behind estimates 4.0 and 2.0 is normal line-up paired back-rank substitutes for like Bishops or Knights either 2 CGs or 2 Gs. Like Cannon in Xiangqi, as Jeremy says, both their values go down as pieces exit. This two-at-a-time back-rank is how 75% CVs introduce new pieces. Not all, just look at Boyer's Grasshopper Chess itself. But that way, staying behind a row of normal Pawns and moving across ranks 1, 2, 3, Contragrasshopper looks pretty effective, unable to be threatened in return very well. Please however continued no CVs using Contragrasshopper is the preferred method of analysis. Let's see, a Contragrasshopper centralized to one of the four d4/5//e/4/5, fully screened, attacks Queen-style 19 squares (Queen's maximum of 27 there minus the 8 adjacent screens). By contrast, Rook threatens only the 14 from any central four, and Grasshopper max is 8 threats once specific screens set. The ratio suggested by those is 5:2, but 4.0 to 2.0-2.5 seems about right for Grasshopper mobility compensation consideration, Charles stresses. Fact is Grasshopper worth gets to about piece range too, would you rather have a Grasshopper or a 2.5-point Camel most circumstances?