Unseen: FJLLLMQWY

Orientation

You and “Opp” (your esteemed opponent, also referred to as “she”) both just played a bingo. It is therefore impossible for either player to infer anything about the other’s rack. The 16 tiles not yet played are randomly distributed between the two racks and the bag. More specifically, you have DEKNOSZ, your opponent has seven of FJLLLMQWY, and the other two tiles are in the bag.

In the analysis that follows, a word that is unacceptable in NWL 2023 but acceptable in CSW 2024 is marked with a pound sign - # - after the word. If the main word is acceptable in both lexicons but at least one of the cross-words it creates is only in CSW-2024, the pound sign is placed in parentheses (after the score if listed). For example, plays at G10 include ZE#, NE, ZEK, NEK#, ZED (#) and NED#, where the cross-play KI is in both lexicons but DI# is CSW-only. If a word is unacceptable in both lexicons, it is flanked by quotation marks (e.g., “MODED”).

Given the stated condition that Opp plays and analyzes perfectly and you trail by 132 points, you cannot win or tie this championship game unless you play a bingo on this turn or the next. For supporting analysis, refer to the extended article (see link at the end).

Hey! There is a playable bingo already on your rack. Is that the answer? ZEDONKS can be played at 15A, hooking CONE with a D (though not at O1, as “MODED” is unacceptable even in CSW 2024). This almost levels the score (489 – 497). It is then Opp’s turn to move, but after that you will receive a substantial going-out bonus (twice the value of the tiles left on Opp’s rack). Isn’t there some possible two-tile combination in the bag (from the remaining pool FJLLLMQWY) with which you can win? Surprisingly, there is not.

If you draw the J, Opp blocks your only J-spot (6F JAB / JIVY) with 5E FY# or MY (if she has the Y) or with the sneaky single-L play of 6G ABLATE. The second tile that you draw (regardless of whether it is F, L, M, Q, W or Y) does not save you, even with your choice of an E or I to play through. For example, neither “JEM” nor “JIM” is a word. Even if Opp had nowhere to play the Q (when in fact, she has three spots), she would win easily. You will eat the J for a humiliating loss. See the left-hand diagram.

Having played ZEDONKS, with NWL 2023 as the lexicon, your best two-tile draw is LL; Opp plays 6F JAB (61), and you go out with N3 ELL (13 points, hooking BACTERIA/L) losing 554-558. See the right-hand diagram.

With CSW 2024, your best draw is FY; Opp plays JAB, and you go out with 2N FY# (26, not shown) and lose 555–558.

If your two-tile draw includes neither two Ls nor the Y, the final score is less close, because Opp can play N2 JELLY (56). This play scores 5 less than JAB but saves her 12 points in rack penalty.

Ironically, ZEDONKS is a kind of self-sabotage. It blocks your own out-play of B13 MY (29#), which would score enough to win.

As bingoing this turn falls short, you need to bingo next turn. Given that there are only two tiles in the bag, and you cannot bingo with fewer than seven tiles on your rack, your viable options are limited to: (a) fishing two tiles; or (b) fishing one tile.

Two-Tile Fishes

When you play two tiles, you replenish them by drawing two tiles, which empties the bag. This means that Opp is endowed not only with perfect playing ability and logic but also with perfect information. She knows exactly what you have on your rack.

Dovetailing your rack with the unseen pool, there are several bingos you can hope for by fishing two tiles:

MONKEYS: You can fish DZ with G10 ZED (16#) or 5A ADZ (13) and draw MY. However, the one spot your bingo plays (O3) can easily be blocked (e.g., by JELL, hooking BACTERIA/L). It can also be outrun by 6F JAB (61) or I9 QIS (43).
DOLMENS: If you fish KZ with G10 ZEK (22) and draw LM, your bingo spots at O1 (97 - hooking MODE/L) or at 15C (91) cannot both be blocked. However, as your fish and bingo are relatively low-scoring, Opp can easily outrun you.
DOWLNES#: Here, again, you fish KZ but this time draw LW and your bingo is threatened at 15C for 94 points. Opp can either outrun it or block it; 14B JELLY (hooking BRONCHIA/L) does both.

The left-hand diagram below helps to illustrate. After playing ZEK, depending on what you draw, you threaten DOLMENS in both the upper right and lower left, or DOWLNES# in the lower left. If Opp responds with JAB as shown (or JELLY), she outruns you. She can also win by choosing to block DOWLNES# if that’s what you have, even with a one-tile play (14C EL or C11 CONEY).

There are additional two-tile fishes:

SKOLLED# (via an NZ fish and LL draw is playable in two spots, indicated by the empty rectangles in the right-hand diagram above. However, your only legal NZ fish, N2 ZEN (34#), is external - it self-blocks one of your spots! Opp loses by 1 point if she tries to outrun 15C SKOLLED# (108) with JAB, but she can win by blocking the lower left (15A FLY scoring the most there) even with the measly one-tile play shown. Neither BACTERIAN# nor BRONCHIAL takes an S-hook.
ENFOLDS or FONDLES also manifests from fishing KZ with G10 ZEK, this time drawing FL. However, you are unable to hook either bingo onto the available squares of O3 or 15C, and SEX interferes with N8 ENFOLDS.
KNOLLED (via an LL draw) bears mention, though you have neither a legal SZ fish nor a spot to play the bingo.

[In an early version of this puzzle, the right-side structure differed: SZ was fishable, and in CONE’s place was TOPE, which takes several relevant hooks (K (#), D, E, and S). KNOLLED and ENFOLDS fit at both 15C and N8/N9, but they could be outrun.]

Aside from Opp playing perfectly, you clearly have several obstacles to surmount:

The initial deficit of 132 points
The limited score of your fishing play
Opp’s high-scoring J-play
Opp’s ability to block if your bingo plays in only one spot

JOLLEYS# (the only J-inclusive bingo in the pool) is unobtainable - you cannot draw JLL with only two tiles in the bag, and anyway there is only one spot to play it. Therefore, if you are able to draw a bingo, Opp has the J. If she can play JELLY for 56 to 59 points (noting, too, a rack-penalty reduction of 28), your fishing play plus bingo need to combine for at least 148 or 151 points. However, if you draw the Y or LL, you limit Opp’s outrunning power to JAB and reduce your fish + bingo burden to “only” 141 points.

For two hypothetical examples of how you might win from a two-tile fish, let’s revisit the right-hand diagram above:

If you had an L in your rack instead of an N, you could play ZEL# (noting that BACTERIAL takes an S, whereas BACTERIAN# does not)
If you had a spot to fish NZ internally for the same 34 points, you would win, as you would threaten a bingo in two spots, and Opp could not block both.

Either way, after her JAB, you could bingo out with SKOLLED# for (at least) 108, plus you would be awarded 52 for the FMNQWY or FLMQWY tiles on Opp’s rack. Let’s do the math: 34 + 108 + 52 = 194, minus 61 from Opp’s JAB, is just enough to overcome your 132-point deficit. Unfortunately for you, a sequence this profitable from a two-tile fish does not exist.

You can score more if your bingo includes a Z (especially if it lands on a DLS square, as well as the word covering a triple line). However, as it happens, from the DEKNOSZ rack, there is no Z-inclusive bingo you can acquire with a two-tile fish. On the other hand, there are two Z-inclusive bingos that you can obtain with a one-tile fish!

One-Tile Fishes

Compared to fishing two tiles, fishing one tile has fewer potential bingo possibilities. On the whole, though, you are more likely to get one of them onto your rack, as for each targeted bingo you are simply trying to draw one specific tile (instead of a parlay of two tiles).

As with two-tile fishes, one-tile fishes can sometimes threaten to play a bingo in two spots; Opp might block one of the threats, but if she is unable to do so with a sufficient score, then you will be able to win with the other threat. One-tile fishes, however, have the advantage of disguise. Because you leave a tile in the bag, Opp has to imagine eight possible racks. She is compelled to contend with both (or all) of the threats she thinks you might have. In other words, one (or more) of your threats can be ”phantom.” If Opp blocks the phantom threat, you can execute your actual threat.

ZONKEYS: You can threaten this 128-point bingo at N8 by fishing your D with 7A OVERTIRED (15) and drawing the Y. If M is left in the bag, you also have a phantom threat of O3 MONKEYS for 111 points. (With a one-tile fish, you cannot get MONKEYS onto your rack, but Opp doesn’t know that.) From her perspective, Opp, who sees a pool of M+ZONKEYS, can outrun both bingos with JAB - beating ZONKEYS by 2 points or MONKEYS by 5 points. See left-hand diagram above.

In case JY is in the bag (instead of MY), you can fish your D with DI# (for a score of 11 instead of 15), grabbing the big Q spot. Opp (denied JAB) cannot outrun ZONKEYS, so she nimbly blocks with 7L PELF. See right-hand diagram above. Opp sees a pool of J+ZONKEYS and knows that from seven of eight racks you might have, you have the muscular JAB but cannot quite win the endgame despite Opp suffering a 12+ point reduction in her Q score. For details, refer to the extended article.

When Opp sees J+ZONKEYS, blocking MONKEYS is unnecessary and incidental. However, PELF (though it takes longer) beats MONKEYS and ZONKEYS by more points than does JAB when Opp sees the pool M+ZONKEYS.

There are three more bingos that you can acquire by making a one-tile fish, two of them having multiple phantom threats. The diagrams below are followed by explanations.

DONZELS: You can fish your K - the best spot being I9 KIS (23) - and draw the L. You threaten to bingo in the two spots denoted by two of the (seven-square) rectangles in the left-hand diagram: DONZELS scores 112 at 15C and 90 at N5. Fortunately for Opp, even if she doesn’t have the J (because it is in the bag), she has the tiles for 14B FELLY (or WELLY), which blocks the higher-scoring bingo spot while outrunning the other.

There are subtleties in this K-fish, L-draw situation: if M or W is the last tile in the bag, then Opp worries not only about DONZELS at 15C and N5, but also about the phantom threat of: (a) DOLMENS at 15C and O1; or (b) 15C DOWLNES#. Regardless, Opp can outrun all bingos with JELLY, JAB, or (usually) FELLY. See again the left-hand diagram.

SNOWKED#: This requires fishing your Z, which you can do for 22 points with ZE# at 14B or N2. If W and Y are both in the bag and you draw either one, Opp sees the tile pool DEKNOSWY and worries about 15C DONKEYS (106) and SNOWKED# at both 15C (105) and O3 (106). Obviously, she cannot block all these bingos but she can outrun them all with JAB or even QIS. (That is, she is fine even if the J is in the bag.) Refer to the right-hand diagram.
DONKEYS: As above (see the same right-hand diagram), you can fish your Z with ZE# at 14B or N2. If you draw the Y and the M is in the bag, Opp sees the tile pool DEKMNOSY and worries about 15C DONKEYS (106) and O3 MONKEYS (111). Here, too, she cannot block both bingos but she can outrun both with the familiar JAB or QIS. Too bad for you!

In summary: By leaving a tile in the bag, you can either (though unfortunately not both): (a) deny Opp the high-scoring J tile, or (b) threaten both a real bingo and an alternate “phantom” bingo. Specifically, in all the above scenarios you might reach containing bifurcated threats -ZONKEYS/MONKEYS, DONZELS/DOLMENS, DONZELS/DOWLNES#, DONKEYS/SNOWKED# and DONKEYS/MONKEYS - no bingo scores enough points; they can all be outrun, even the 128-pointer. There is one more lever that you need to apply.

Key Threat

The first critical observation is that you can fish your Z with H11 DITZ. It might seem crazy to try such a setup when the Y is most likely on Opp’s rack, but these are desperate times; you must hope that Y is one of the two tiles in the bag and that you will draw it! H11 DITZ scores only 14 points but sets up a Y-hook that adds 54 to the score otherwise achieved by a pending bingo. Without the coveted Y-hook, 15C DONKEYS scores 106. With the coveted Y-hook, DONKEYS scores 160 and cannot be outrun by JAB, even accounting for the points you sacrifice by fishing the Z on H14 (instead of on 14B or N2).

You may well inquire as to why Opp cannot simply block DONKEYS. She can, but she might not choose to if … M is in the bag! This is the only possible unseen tile from her point of view whereby you might be holding one of two different bingos and she can defeat either one but not both. She can reason thusly: “Currently, D+MONKEYS are the tiles unseen to me. If my opponent held any six of EKMNOSY when he [referring to you] fished the Z, he could have drawn the tile he needed for MONKEYS and left the D in the bag. If I block the bottom with 14B JELL - my highest score there, he will go out with O3 MONKEYS and tie the game at 532–532!”

It is nevertheless true that after H11 DITZ, if Opp blocks MONKEYS and you play DONKEYS (cashing the monster Y-hook), you win! Given that your winning is a bigger threat to Opp than your tying, why wouldn’t she block the bottom anyway?

This brings us to the second critical observation: you can fish your Z in the other direction. If Opp always blocks the bottom after H11 DITZ, this suggests that the better placement for you is 8K DITZ (on the right), pretending to be trying to win with MONKEYS. That way, when Opp falls for your misdirection, DONKEYS will end the game in a 529–529 tie. After all, the ploy cannot work unless M is in the bag (and you draw the Y you need), the symmetrical scenario being this: Opp thinks that D might be in the bag, noting that she can see neither the D nor the M. The only threats that she can envision are – equally - DONKEYS and MONKEYS.

For a visual clarification of Opp’s plight, examine the graphic aid below.

Minimax Equilibrium

So, does that mean that you should play 8K DITZ, expecting Opp to block MONKEYS, so that you always tie with DONKEYS? If that’s the way it worked out, then yes - that’s a good deal for you. However, Opp can do better! She can adopt a randomized or mixed strategy by straightforwardly blocking two-thirds of the time and “second guessing” - calling what might be your bluff - one-third of the time. (There are ways to do this without a physical aid, but if she has an analog wristwatch, she can decide that if the second-hand is on the first third of the dial she will second guess and otherwise she will block- then glance at the dial and obey.)

For the purposes of the following discussion, we will count a win as +1, a loss as –1 and a tie as zero. (Other scoring systems are possible, but this scoring system helps clarify why game theorists regard Scrabble as a zero-sum game.)

Suppose that you always play DITZ at 8K, on the right. Opp will block 2/3 of the time and tie those games (when you play 15C DONKEYS without a Y-hook); those tying occurrences count zero. The other 1/3 of the time, she will second guess your feint to the right and block the bottom, winning those games (worth +1). Her (average) “equity,” then, is (2/3 × 0) + (1/3 × 1) = +1/3.

What happens when you instead play DITZ at the bottom? Adopting the same strategy (being agnostic as to whether you might be holding DONKEYS or MONKEYS), Opp blocks the lower left 2/3 of the time (outright winning those games), and second guesses you by blocking the upper right 1/3 of the time (losing those games to the big DONKEYS play). She nets (2/3 × 1) – (1/3 × 1) = +1/3 (again). In short, with this mixed strategy, she will net exactly the same equity whether the DITZ you try is on the right or at the bottom. As our scoring system is zero-sum, your equity is (of course) –1/3, the negative of Opp’s +1/3.

If you yourself adopt a balanced strategy, you bluff with 8K DITZ with 2/3 frequency and straightforwardly set up H11 DITZ with only 1/3 frequency. Suppose, for example, Opp always blocks. Your 8K DITZ played 2/3 of the time ties, and your H11 DITZ played 1/3 of the time loses; that comes to (2/3 × 0) – (1/3 × 1) = –1/3. Conversely, if Opp always second guesses, your 8K DITZ played 2/3 loses and your H11 DITZ played 1/3 wins; that comes to –2/3 + 1/3 = –1/3. Your equity is –1/3 either way.

Stated formally, there is a key theorem about two-player zero-sum games called the "minimax theorem," according to which there exists a mixed strategy for Player 1 and a mixed strategy for Player 2 that together form an "equilibrium," meaning that neither player can profit by deviating unilaterally.

Indeed, if Opp’s mixed strategy is faithfully implemented in the manner described above, her equity is +1/3 and your equity is –1/3, regardless of what mixed strategy (40:60, 90:10, reading tea leaves, whatever) that you might care to adopt. Likewise, if your mixed strategy is implemented as above, the respective equities are the same regardless of what mixed strategy that Opp might care to adopt.

The four diagrams below provide visual confirmation of the pivotal sequences.

In summary, the sequence can play out in four ways, as the two diagrams above and the two diagrams below illustrate. As indicated in the table just below, 4/9 of the time you tie—your equity is zero. The four sequences sum to: 0 – 2/9 – 2/9 + 1/9 = –3/9, which is -1/3. (Your opponent’s equity is the opposite: +1/3.)

Assuming that in playing DITZ, you are fortunate enough to draw the Y and M is in the bag, then … With either or both players adopting the balanced [2/3, 1/3] mixed strategy, you have a 4/9 chance to tie, a 4/9 chance to lose, and a 1/9 chance to win. In dollar terms: suppose that first prize is $12,000 and second prize $6000; on average, Opp will take home $10,000 and you will take home $8000 (1/3 of the way between $9000 if you tie and $6000 if you lose).

To result in equities different than +1/3 for Opp and –1/3 for you, it is necessary for both players to deviate from the recommended mixed strategy. As an example, consider a strategy where you always bluff and Opp always blocks. We looked at this before when you misdirected Opp with 8K DITZ, but if she is truly D/M agnostic, then she will always block regardless of DITZ direction. Given the actual bingo on your rack, where 8K DITZ results in your always tying with DONKEYS, it attains an equity of zero, which is better than the –1/3 produced from a mixed strategy. On the other hand, if Opp is somehow on to you and always assumes you are bluffing, then your equity plummets to –1 (you always lose). In short, if both players deviate, there exist both risk and reward.

Because the equity difference between a win and a loss is twice the difference between a win and a tie, the act of both players applying the aforementioned [2/3, 1/3] mixed strategy in this situation strikes the equilibrium. In game theory, honoring this equilibrium is deemed to be the “best” strategy for both sides. It is the only strategy that guarantees a certain minimal equity: in this case, +1/3 for Opp, –1/3 for you. In any event, from the standpoint of equity, it is clearly the safest strategy!

Accordingly, we promote the answer to this puzzle as the best play being a mixed strategy. Holding DEKNOSZ, you should play 8K DITZ with 2/3 frequency and H11 DITZ with 1/3 frequency. (If your rack was MEKNOSZ, you should do the opposite.) For Opp’s part, assuming that you are lucky enough that M is in the bag (otherwise she holds the M and knows that MONKEYS is no threat, so she blocks DONKEYS and wins), she should block your DITZ misdirection with 2/3 frequency and second guess with 1/3 frequency.

For a more mathematically oriented version of this article, focusing to a greater extent on mixed strategies and minimax equilibria, read our paper Bluffing in Scrabble at https://timothychow.net/cv.html.

Wrap-up

Two situations are worth revisiting. They would have been covered in the One-tile Fishes section, except that it was too early to reveal DITZ as a way of fishing the Z.

If you play 8K DITZ, draw Y, and leave J in the bag, Opp has the M on her rack and need not block MONKEYS. Still, seeing the pool J+DONKEYS, she is threatened with both 15C DONKEYS (106) and O4 JOKEY (135). The latter scores 29 more points but is the lesser threat because it doesn’t go out. Opp can outrun both threats with QIS, or she can block DONKEYS with a decent score in the lower left, intending QIS (or JAB if she draws the J) next turn. Refer to the extended article.
If you play DITZ, draw W, and leave Y in the bag, Opp worries about DONKEYS in the bottom left and SNOWKED# on one side only (the Z self-blocks your other spot). If you play 8K DITZ, Opp blocks the lower left and you have nowhere to play either bingo. See left-hand diagram below. If you play H11 DITZ, it gets more exciting, but (regardless) Opp blocks the giant DONKEYS spot with 14B JELL, and if you bingo out with O3 SNOWKED# (106), you lose by 3 points. See right-hand diagram. Casting drama aside and remembering that point spread is irrelevant, you should entirely discount the possibility of WY being in the bag. You still play 8K DITZ 2/3 of the time and H11 DITZ 1/3 of the time, in case MY is in the bag.

The point is that in this very similar DONKEYS/other dual-threat scenario, Opp has no incentive to block O3 SNOWKED#. It scores 3 fewer points than MONKEYS (rack-penalty adjusted), which makes the difference between you losing and you tying.

This puzzle works materially the same way (with the featured position and tile pool) for any rack that has five tiles out of EKNOSY: (a) plus DZ or (b) plus MZ. We tested all 12 of these racks (six with D and six with M) to ascertain that there is no move that has a chance to win or tie other than DITZ. This detailed cross-check informs us that there are 12 different sound versions of this puzzle!

More importantly, though, this cross-check verifies that the [2/3, 1/3] equilibrium does not need to be adjusted. This is the case even though it is possible to fish for additional bingos (MENFOLK, ZEDONKS, DOWLNEY#, MYELONS#, SMOYLED# and the eight-letter SOLEMNLY) from one or more of the other 11 racks, and from one of those it is even possible to win (if CSW 2024 is used) more than one third of the time. For details, see the extended article.

There is one neutral rack: EKNOSYZ. We call it the “13^th warrior.” From this rack, when you play DITZ and DM is (necessarily) in the bag, you do not know which of D or M you will draw, and therefore you have no directional bias for DITZ when you play it. To avoid having to slightly adjust the [2/3, 1/3] equilibrium (see the extended article), we made EKNOSYZ the only one of the 13 racks that has a play with better equity than DITZ. You should bingo with N8 ZONKEYS, slightly overcoming Opp’s JAB or 14B FJELD. If you draw DL (1/12 chance) you win 560-558, or if you draw LM (another 1/12 chance), you win 543-542.

With the inclusion of a 128-point spot on the board for ZONKEYS, it is crucial that none of the 12 relevant racks can successfully fish for that bingo. Indeed, none can. Refer to the extended article for details.

In this puzzle, your game equity is poor, no doubt. You have a 1/72 chance of drawing Y with M in the bag, and your equity is –1/3 in that case. However, you walk away with the pride of having made an ingenious and unprecedented play and just maybe first prize!

Prior to reading the solution, if you contemplated DITZ as a serious candidate even though Opp probably has the Y, kudos. If you realized that you need both to draw Y and for M to be in the bag, then super-kudos! If, further, you pondered bluffing and second guessing and figured out that you have a chance to win (not just tie) with DITZ, then please let us know - we will be profoundly impressed!

For complete two-tile fish and one-tile fish charts, more detailed logic and analysis, endgame variations, and various extravaganzas, refer to the extended article at: https://timothychow.net/EquilibriumPuzzle_extended.pdf.

If you would like to kindly offer comments on any of our articles, feel free to e-mail us at one or both of these addresses:

We acknowledge Joe Edley and Jerry Lerman (who designed the graphic) and Charlie Carroll. Their comments and suggestions for improving the presentation of the article were very helpful.

About the authors:

NICK BALLARD, affectionately known as “Nack,” is an expert of many games, including Chess, Go, Color Lines and Scrabble (rated 2026). He is currently writing a seven-volume series: Backgammon Openings: Early Doubles.
TIMOTHY Y. CHOW received his Ph.D. in mathematics from MIT in 1995 and works at the Center for Communications Research in Princeton. His research interests include algebraic combinatorics and computational complexity theory.

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