Check out Glinski's Hexagonal Chess, our featured variant for May, 2024.

Worse than Worthless

By Ralph Betza

Something that is worthless has zero value. What could be worse?

Negative Money

My dentist told me the fee for today's visit was zero, and he hoped I had that amount with me. Ha ha.

I immediately went into an uncommunicative reverie, thinking about negative-valued money. If I owe you ten dollars, you can give me a minus-ten dollar bill.

There would need to be severe penalties against destroying or discarding or even hiding negative money, to prevent its abuse. Imagine the anti-miser, who lives well beyond his apparent means and after death is discovered to have a huge stash of negative money hidden in mattress.

Strangely enough, negative money has some positive value. If you have a debt, you pay interest on it, but if you have negative cash, it is like having an interest-free loan. Therefore negative money would be somewhat sought after, as people would try to keep all their liabilities in uncash.

Counterfeit negative money would not be a problem.

The Negative Relay Knight

In the mid 1970s, I did an article for NOST-algia in which I explored "all possible" variants of Relay Chess. One such variant allows only your opponent to benefit from the relay powers.

Consider a piece that moves as a Knight but has no capturing powers; instead, any enemy piece a Knight's move away from it gains temporarily the power of moving or capturing as a Knight. This is a piece of negative value!

Following Mannis Charosh's rules, the enemy King cannot benefit from the relay power, and the enemy Pawns cannot use this power to move to their first or eighth ranks. However, I will break from the Charosh tradition and allow the Negative Relay Knight to give an enemy Negative Relay Knight the temporary power to move or capture as N, and my reason for doing so is that I have seen that the resulting tactics are piquant and charming.

Using the above rules, and replacing each side's Knights with Negative Relay Knights, we get a simple game called Negative Relay Chess. Here is a sample opening:

1. NegRelayNb1-c3 d7-d5 2. NegRelayNc3-e4 e7-e5 3. NegRelayN e4-f6+ NegRelayNg8-h6 4. NegRelayNf6-g8; and White has succeeded in burying his negative-valued piece deep in enemy territory where it blocks enemy moves and where the benefits the foe gains from it are less dangerous (but where the opponent is more likely to be able to get pieces in range of it). This is a strategy that might be worthless or worse, and to achieve it W has lost several tempi.

Meanwhile, the pieces at h6 and g8 temporarily give each other the ability to capture. Notice that the Negative Relay Knight can be captured, but can be used as though it were uncapturable because, due to its negative value, capturing it is usually undesirable.

The move 3.Ne4-f6+ tries to force the capture of the liability. Instead, Black's reply 3...Ng8-h6 provides an illustration of the temporary nature of the relay power. The sample game is very short, but provides insight into the rules, the strategy, and the tactics; I think it's a pretty good sample game!

There is absolutely no doubt that one can play Negative Knight Relay Chess with Different Armies. Beware confusion and danger of illegal moves when negative Knight and negative Fibnif mutually attack.

What's it Worth?

The Negative Relay Knight has some small potential positive value, because it could be used to blockade a passed Pawn or to make tempo moves in the late endgame. In fact, in the late endgame, because there are so few enemy pieces to be helped by it, it might even have a very small positive value overall.

Its whole-game value can be estimated as the amount of average mobility it gives to the enemy times the chance that it does so, and the chance that it does so is its own average mobility times half the probability that a random square is occupied. That's -4.59375, and since the Knight's positive average mobility is 5.25, replacing a Knight with a Negative Relay N makes your army weaker by 9.8, or one and seven-eighths of a Knight; in other words, replacing a pair of N with Negative Relay Knights should theoretically make your army 11.25 Pawns weaker!

Because the N starts at g1 and b1, where its friendly Pawns shield it from undesired contact with enemy pieces, an army that has 3.75 Knights' worth of power distributed among its other pieces might be able to simply win with its other pieces while leaving its negative-valued Knights at home, right? The positive value of having a worse than worthless piece in your army is that you get to have so much power added to your other pieces!

Maybe not. With the funny material balance, the levelling effect will hurt the pieces that have added powers -- for example if you have Cardinals instead of Bishops, the enemy Bishops gain the ability to chase your Cardinals away.

In order to keep the strategy of keeping the negatives at home from being too strong, I'm going to avoid giving jumping moves as enhancements to any of the pieces. This means that the negative pieces, if left at home, will have the added liability of being in the way of smooth development. Also, because the army with negative pieces has to move Pawns to develop, it must break up the Pawn formation that shields the liabilities from contact with the foe.

Let's let the Rook have the additional power of moving as a three-square limited Bishop (that is, as a B3 that can't move farther than from a1 to d4), and the Bishop also move as R2 (making normal Bishop moves or short Rook moves, for example f1 to f3 but no further).

I haven't worked out the exact arithmetic of that, but it's close enough to be worth a try. It feels right, too. But omigosh, what an experimental army, how exotic it is, and oh, so great a chance that it is too weak or too strong! My brain is tired and I can't playtest it blindfold; but it's such an exciting idea, and such a crazy army, I can't resist publishing it as is, just eight hours after the dentist's joke, with apologies if it needs to be corrected later.

The Nattering Nabobs of Negativity

The Nattering Nabobs of Negativity[1] are a highly experimantal and untested army intended to fit into the framework of Chess with Different Armies.

The Queen, King, and Pawns are the normal FIDE Q, K, and P.

The R is the FIDE Rook plus short bishop of length 3 -- RB3 -- and because it ought to have a name I'll call it a Rhubarb.

The B is the FIDE Bishop plus short Rook of length 2 -- R2B -- Rutabaga is its name.

The Knight is a Negative Relay Knight, as described previously in this document. It's called the Ruthven, named after the Murgatroyd whose family curse made his title of nobility worse than worthless.


I don't have the energy to discuss the following ideas in complete detail.

Asymmetrical Relay

"Its whole-game value can be estimated as the amount of average mobility it gives to the enemy times the chance that it does so, and the chance that it does so is its own average mobility times half the probability that a random square is occupied." Perhaps you can detect that my particular way of phrasing that statement encompasses the possibility of a piece moving as Knight, and granting any piece a King's move away from it the right to move as a Rook or to capture as a Bishop?

Promiscuous Negative Relay

Notice that friendly pieces are not affected by the relay power of a friendly piece. Unfortunately, the opening position is so crowded that if a friendly Pawn a Knight's move away from a friendly Knight gained the liability of being used by the enemy to move or capture as a Knight, chaos might ensue. However, since Pawns may not move to their own first rank, the game might be playable? But if Pd2 captures Bf1 and Bf1 is also a liability, then the capture would be undesirable.

Therefore a game in which all pieces (except perhaps King) are negative relay pieces both to friend and foe might be playable after all!

Huge Negative Values

"Its whole-game value can be estimated as the amount of average mobility it gives to the enemy times the chance that it does so, and the chance that it does so is its own average mobility times half the probability that a random square is occupied." For a symmetrical relay piece, that is, in the natural case where the power of movement is the same as the power granted, the negative value varies as the square of the average mobility of the power. In other words, the negative relay Amazon would have in theory sixteen times the negative value of the negative relay Knight! Sixteen times!

That's more negative value in one piece than the positive value of an entire normal army of pieces!

[1] Phrase coined by William Safire for a speech by Spiro T. Agnew.

Written by Ralph Betza.
WWW page created: November 6th, 2001.