My friend and I were playing chess the other day, and while he was admiring some birds in the park I pulled a fast one and stuck one of my pieces back on the board. Unfortunately, my new-found criminal career was doomed from the start, for he quickly noticed my deception.
“Did you see me out of the corner of your eye?” I asked.
“Yes,” my friend admitted, “but to tell you the truth, I would have known even if I was a complete stranger just passing by.”
I laughed. “Don’t be silly! A stranger wouldn’t even know who’s turn it is, let alone that there is an extra piece on the board.”
“It’s the power of logic, my friend,” he said with a smile. He then proceeded to embark upon the most fascinating journey of deduction that I have ever heard. Twists and turns abounded at every corner! And, at the end, he managed to prove with mathematical precision exactly which piece on the board should, in fact, not have been there.
Before I continue with the story, I urge you to look at the board below and see if you can figure out which piece is the extra piece. Do not assume that my friend and I played smart chess, only that we followed the rules. Also assume that there is only the one error on the board.
“Ok,” I said, “Prove it. Who’s turn is it?”
“Must you start off so easy?” said my friend. “White is in check, therefor black must have moved last.”
I took a quick glance at the board and conceded the point. “Ok, but that doesn’t mean there is an extra piece.”
“Look at the bishop on c3. How did it get there?”
I saw that the bishop was a dark-squared bishop that should have come from c1. “The pawns on d2 and b2 haven’t moved, so the bishop could not have gotten to c3. Dear heavens, did we screw up our game?”
“Not at all. The bishop on c3 must be a promoted bishop. It is the only explanation. White must have promoted one of his pawns to a bishop.”
“There is another possibility,” I said. “Maybe we just screwed up and the bishop shouldn’t be there.”
“No,” said my friend, “the bishop is needed because without it the white king is in check from two places at once.”
“Oh. Never mind. I still do not see why there must be an extra piece.”
“We know that white promoted a pawn to a bishop, but there are 8 white pawns on the board. Thus, one of the white pawns should not be there.”
I felt a smile creeping to my face. Perhaps my friend was on to something after all. “Go on…”
“The extra pawn can’t be b2 or d2 for the same reason the extra piece couldn’t be the bishop; without those two pawns the white king is in check from two places at once.”
“I see!” I exclaimed. “We have proven the existence of an extra piece, and we have narrowed down our list of suspects to six pawns. But what now? None of the other pawns are needed.”
“It’s true that we cannot deduce much more about the white pawns, so let’s focus on the other pieces. There are three white pieces not accounted for: the two original bishops and one of the knights. Can you tell me how many of these white pieces were captured by black pawns?”
I thought really hard. “Well, the black pawns are all on their original files—”
“Or the g and h pawns could have switched places!”
“—right. Or those two pawns switched places. Since there are only three white pieces missing, the black pawns must have captured either 0 or 2 pieces. An odd number of captured pieces would have left two black pawns on the same file.”
“You are almost there,” said my friend. “Which is it? 0 or 2?”
“I have no idea,” I admitted.
“No worries. Let’s take this one piece at a time. Did a pawn take the original white bishop on c1?”
“No! The white bishop on c1 never moved from that square because it’s locked in by the pawns on b2 and d2.”
“Good! Now, what about the bishop from f1? Did a pawn take it?”
“No!” I exclaimed. “That bishop is also locked in by pawns.”
“Be careful,” said my friend. “The pawns on b2 and d2 are necessary to keep the king from being in check from two places at once. However, the pawns that are locking in the white bishop from f1 are not necessary. Either one of e2 or g2 could be the extra piece that shouldn’t be on the board. If that is the case, the white bishop could have moved to another part of the board where it was taken by a black pawn.”
“Oh,” I said, very sadly. “So, what gives?”
“Let’s go back to the very beginning of our deduction. Black just moved his rook to put white into check. Where did his rook move from?”
“The situation seems almost impossible. The only possibility is that black’s rook took a white piece on f1 after moving from either g1 or h1.” Suddenly, it came to me (or so I thought). “Black’s rook must have taken the white bishop on f1!”
“Again, be careful,” my friend warned. “If white’s bishop was on f1, how did the black rook get to g1 or h1 in the first place? And, for that matter, how did the white rook get out?”
“Well, I suppose that means the g2 or h2 pawn must be the extra piece, because there had to have been a hole for the rooks to get through.”
“I’ll say it once more, be careful. Try out this scenario: The e2 pawn is the extra piece. At some point in the game, the f1 bishop left its home square. At another point in the game, the black rook moved to g1 or h1. Finally, at a third point in the game, the black rook was blocked by a white knight. In this scenario, the last move was the black rook taking the white knight.”
“Ok, ok,” I moaned, “but it doesn’t matter. In any case, the black rook took something. Maybe it took the white knight, and maybe it took the white bishop. But it took something.”
“Not necessarily. There is one more possibility. What if the f2 pawn is the extra piece? In that case, the rook could have simply moved to f1 from anywhere along the f file. No capturing is necessary.”
“^*$&!” I exclaimed.
“Don’t worry, Phil. It all works out nicely in the end. In this final scenario, with the f2 pawn being the extra piece, what do we know about the f1 white bishop?”
I quickly came to my senses. “We know it never left its home square. If the f2 pawn should not be there, then the e2 and g2 pawns should be, thus locking in the white bishop.”
“Exactly! In that scenario, nobody knows what happened to the white bishop from f1. All we know is that it never left its home square and thus could not have been captured by a pawn.”
“Whew,” I breathed. “Can you sum that up for me?”
“Sure. When you boil it all down, here is what we have:
- Three white pieces are missing.
- The black pawns captured either 0 or 2 of them, because an odd number of captures would leave two black pawns on the same file.
- The c1 bishop was not taken by a black pawn, because it never left its home square.
- Some other white piece was captured on f1. It could have been a piece captured by the rook moving to check the king. Or, the other white bishop was trapped there by e2 and g2, in which case it never left its home square and must have been captured there.
- Thus, two of the three captured white pieces were not captured by a black pawn. That leaves only one piece that could have possibly been captured by a black pawn.
- Conclusion: Since the black pawns captured either 0 or 2 pieces, and they also captured at most 1 piece, they must have captured 0 pieces.”
My friend finished speaking and I could feel my head spinning. “Wow,” I said. “All that trouble only to prove that no black pawn did any capturing. What good does that do us?”
“What good does it do?” said my friend. “Why, it’s only the key to the entire mystery!”
“Then please share it.”
“If none of the black pawns moved off their home file, how did a white pawn get through to promote to the c3 bishop?”
“One of the white pawns must have moved diagonally by making a capture.”
“And how many black pieces are missing?”
I saw where my friend was going. “Only one black piece is missing! Therefor, one of white’s pawns moved exactly one diagonal in order to move around a black pawn and make it to the 8th rank.”
“So, Philip, how does that narrow our list of suspects?”
“We already know that it wasn’t pawns b2 or d2. Now we know it must have been a pawn that could only make one diagonal move. That rules out a2, because it would have to make two captures to get around a5. It rules out c6, because it would have to make two captures to get to either the a file or the e file. It rules out e2 because it would need two captures to get around e5.”
“What is left?”
“There are now only three possibilities: the f2 pawn promoted on e8 or g8, or the g2 pawn promoted on h8, or the h2 pawn promoted on g8. We are getting really close!”
“Very close indeed, Philip. There are three possible squares that the pawn could have promoted on: e8, g8, and h8. What do you notice about the color of those squares and the color of the promoted bishop?”
I literally gasped as the final piece of the puzzle fell into place. “The promoted bishop is a dark-squared bishop. If the pawn had promoted on e8 or g8, it would have been a light-squared bishop. The only pawn that could have reached h8 with only one diagonal move, and thus could have promoted to a dark-squared bishop, is the g2 pawn!!!! The mystery is solved! The g2 pawn should not be on the board because it was promoted to the bishop on c3!”
My friend got up to shake my hand. “Well done, Philip. I knew you had it in you. Now, remove that nasty pawn and let me get back to whipping your butt. It’s your turn. Mate in…”