You are about to play a game of chess with the ghost of Bobby Fisher (at age 24). Bobby has agreed to a concession: In a pre-game round, you may select any one of your pawns and activate it by advancing F₂ squares (here F₂ = second Fibonacci number = 1). Next, you activate the pawn located F₃=2 columns to the right, moving it F₄ squares forward, and so on. (Each pawn always stay within their native column.)

If at some point you return to a pawn which was previously moved, you must then move it in the opposite sense. That is, any given pawn will alternatively move forward ±F_2M rows (where M is the appropriate turn counter and a negative move corresponds to a move backward).

Whichever of Bobby's pieces (except the king) you land on with one of your pawns are removed from the board. If at some point you were to land on the square occupied by the Bobby's king, or by one of your own pieces, the pawn simply shares that square without a capture/removal.

You may continue this procedure until such time as you are no longer able to capture any further of Bobby's pieces. After these preliminaries, you return your pawns to their normal positions and Bobby will play white against you, using whatever pieces still remain to him. Bobby congratulates himself on having granting you a generous advantage.

How generous was Bobby and which pawn should you activate first?

Clarification and example: Wraparound is used to deal with “off the board” locations. For instance, if the formula requires you to select a pawn one column beyond the right edge of the board, you should revert back to the leftmost column.

An example may help clarify the procedure. After ten times moving pawns in this way, you would next activate the pawn located F₂₁ columns to the right of the most recently moved pawn. Since F₂₁=10946, we determine (using wraparound) that the pawn to activate lies two columns to the right (alternatively, six columns to the left) of the most recently activated pawn. Suppose for the sake of illustration that this pawn is currently positioned in the fifth row, having been moved exactly once previously. We now have to move it back by F₂₂=17711 rows which, using wraparound, would reposition it into the sixth row.

From a quote by Grandmaster Bobby Fischer, himself, "Concentrate on material gains. Whatever your opponent gives you take, unless you see a good reason not to.", we see we should probably take his advice. But which pawn?

Mary Elizabeth Braddon, in her 1868 book Belgravia, wrote: "...the removal of the king's pawn is not an unalloyed loss to the giver of the odds....The sacrifice requires a move, which is saved when the pawn is given...." If we take her advice, then, we should ignore e2 as the pawn of choice as it is not considered a handicap by the masters.

Reading the analysis of those that have studied the game in giving odds, such as Larry Kaufman, author of the 2006 online article Giving Odds: Pawn and Move(s), we see how an expert rates the pawn handicaps usually played. The c2 pawn is considered the weakest. A little stronger is the a2 pawn. But the pawn considered the strongest handicap is the f-pawn. The f7 pawn and move is the classical handicap given when a single pawn was offered. When only material was offered, the pawn usually given was the f2 pawn.

Sometimes a Grandmaster would play with even greater odds. Quoting from Kaufman's article, "White's two-pawn advantage is not much less than knight odds, so this handicap is rare, though Fischer is said to have played it against masters." 

So to answer the first half of the question, "How generous was Bobby...?"  As far has handicaps go, the single pawn is just above minimal as he allowed us to choose which single pawn to remove. This leads us to answer the second half of the question: "...and which pawn should you activate first?"  As noted by Charlie, the single pawn that makes the capture is the pawn four to the right or left of the original pawn moved. Since we wish to garnish the greatest handicap, to locate the capturing pawn in the f column, our initial pawn should be b7.

Edited on April 6, 2008, 11:28 am

Posted by Dej Mar on 2008-04-06 11:16:11