Endgame analysis (warning: can be mathematical)

Here’s the position again:


(click to enlarge)

Get familiar with the position if you’re going to read through the whole thing here, you’ll probably need to refer to it quite a few times.

Some things to note:
1. Tony has just bingoed, so it’s impossible to tell what he drew from the bag.
2. There are at least 6 viable bingo spots, and it’s impossible to block all with 1 move here.

So what’s the winning-est move here for Ricky? In this case, the best thing to do would be to fish for a bingo, while hoping that Tony doesn’t get a bingo the next move.

These are the possible bingos for Tony:

15D ROTTENLY 65
15H TROTLINE 77
15H TOILETRY 86

O7 TORTILE 88
O7 LOTTERY 97

J6 ELYTROID 66
J6 INTORTED 63

J2 ENTIRELY 63
J2 ENTIRETY 63
J2 ETERNITY 63

K4 ELYTRON 72
K4 INERTLY 72

10H NITROSYL 71

Bingo-able racks are: ELNORTY, EILNRTY, EINRTTY, EILORTY, EINORTT, ELORTTY, EILNORT, EILORTT, ILNORTY – that makes 9 racks out of a possible 18 racks. This means that if Ricky empties the bag, Tony may be able to bingo no matter where Ricky plays.

Ricky’s rack, however, is not very promising itself. Let’s look at the possible options.

ONE TILE FISH

These are the possible bingos he may pick up with a one-tile fish:

7 letter, K4 (all 64 points)
AIERIES
AUDITEE
EQUINIA
EUCAINE
EUGARIE
EUTEXIA

8 letter, to the D:
AERIFIED (60)
AURIFIED (60)
BEAUTIED (60)
UNIDEAED (62)

8 letter, to the T:
EUTAXITE (58)
INERTIAE (58)
METAIRIE (58)
MINUTIAE (57)
UINTAITE (58)

Let’s sort these into the specific racks:
?AEEII
Draw R – AIERIES, AERIFIED, INERTIAE, METAIRIE

?AEEIU
Draw N – EUCAINE, UNIDEAED
Draw R – EUGARIE
Draw T – AUDITEE, EUTEXIA, EUTAXITE, BEAUTIED

?AEIIU
Draw N – EQUINIA, MINUTIAE, UINTAITE
Draw R – AURIFIED
Draw T – UINTAITE

I suppose we can rule out solely playing off the U as it would mean a 1/9 chance of bingo for Ricky. Playing off the extra E or I would mean a 3/9 chance of picking a bingo.

Now between playing off the E and the I, I would argue that it is better to play the I off. If Ricky fished the I off, either an N or T pick would give him at least two spots for bingo. If he fished the E off and drew a T, Tony could block off Ricky’s only bingo spot. The prime fishing spot for the I is at O1, (B)I/I(N) for 18 points.

So from here, there are 4 possible scenarios (let the event of Tony playing a bingo be A, therefore the event of Tony not playing a bingo is A’, at the same time let the event of Ricky drawing a playable bingo after the fish be B, therefore the event of Ricky not drawing a playable bingo is B’):

1. A & B

Probability of this happening is 1/18*1/2 + 1/18*1/2 + 1/18*0 + 1/18*1 + 1/18*0 + 1/18*1/2 + 1/18*1/2 + 1/18*1/2 + 1/18*1/2 = 2/9

What I did, basically, was to find the probability that Tony has a single rack – 1/18, multiply it by the probability that Ricky draws a bingo, knowing which 2 tiles are left, and adding it all up for all the possible bingo racks Tony has (following the sequence listed above). Ask me if you need further explanation.

There appears to be three cases where Tony bingos and wins, if Ricky does fish at O1 and reply with his own bingo.

a. Ricky plays off I and draws N, score 391-410
b. Tony plays LOTTERY for 97, score 488-410
c. Ricky plays EUcAINE for 64, score 488-474
d. +2 for the I, 488-476

a. Ricky plays off I and draws N, score 391-410
b. Tony plays TOILETRY for 86, score 477-410
c. Ricky plays EUcAINE, for 64, score 477-474
d. +2 for the T, 477-476

a. Ricky plays off I and draws T, score 391-410
b. Tony plays TOILETRY for 86, score 477-410
c. Ricky plays AUdITEE or EUTExIA, for 64, score 477-474
d. +2 for the N, 477-476

This means there’s a 2/9 chance of A and B happening, and Ricky will lose 1/12 out of these 2/9 – win probability from number 1 is 11/54.

2. A & B’

OK the probability of this happening is easy to calculate – just 1/2 – 2/9 = 5/18

Ricky will lose if this happens, for obvious reasons.

3. A’ & B

Non-bingo racks for Tony are EILNORY, ELNORTT, ILORTTY, EIORTTY, EINORTY, INORTTY, EILOTTY, EINOTTY

Probability of this happening is, using a similar mathematical method as above, 1/18*1 + 1/18*0 + 1/18*1/2 + 1/18*1/2 + 1/18*1/2 + 1/18*0 + 1/18*1 (this is complicated as EUGArIE can be blocked, but Tony would lose anyway) + 1/18*1/2 = 2/9

Ricky will win this if this happens.

4. A’ & B’

Probability of this happening is 5/18, using the same method as in number 2.

This is probably the hardest situation to analyse, but I did it by going the possibilities in Quackle, using Championship Player/ simulation when necessary (yes, I really did). As it is, Tony will win all situations here because of the TWS at O8.

Hence, total win probability of playing (B)I/I(N) O1 for 18 points is 11/54 + 2/9 = 23/54.

TWO TILE FISH

A few things to consider here:

1. Where to fish

This has some impact on the outcome because Ricky’s fish could block Tony’s bingo or his own bingo, though there is always a possibility that Tony bingos somewhere else.

2. What tiles to fish off

This affects the probabilities that Ricky will draw a bingo on his next move.

To analyse this we have to list Tony’s possible bingos by the racks:

ELNORTY
ELYTRON
ROTTENLY
Unblockable

EILNRTY
ENTIRELY
INERTLY
Blockable

EINRTTY
ENTIRETY
ETERNITY
Blockable

EILORTY
ELYTROID
TOILETRY
Blockable

EINORTT
INTORTED
Blockable

ELORTTY
LOTTERY
Blockable

EILNORT
TROTLINE
Blockable

EILORTT
TORTILE
Blockable

ILNORTY
NITROSYL
Blockable

Where can Ricky play his block? Either at 10M to the S, 12M to the N, J13 to the D or J3 to the E (probably EAU then). There’s no point playing at the T at 15H because playing at J13 blocks an additional line.

What’s the best block, then? According to the racks above, blocking J13 and 10M seem to be the best – they block 3 possible bingo racks each. Comparatively, playing at J3 or 12M blocks only 2 possible bingo racks.

As for the vowels picked, it is very difficult to count the number of possible bingos played (and I won’t do so) – I’ll just generate a list of the 4 vowels he can keep, plus 3 blanks and see how many bingos there are, for an approximate guide.

AEEI – 168
AEII – 88
AEIU – 160
AIIU – 24
EEIU – 70
EEII – 49
EIIU – 31

From the draw rates, it seems that the probability of Ricky playing a bingo (if Tony doesn’t bingo out) is very high, probably about 80%? (NOTE: this is an arbitrary estimate – would be good if someone calculates this, but it seems very hard)

Probability of Ricky winning from this = P(T0ny not playing a bingo)*P(Ricky playing a bingo) ≈ 2/3*80% = 53.33%

OTHERS

This means either passing or playing more than 2 tiles – quite obviously the former is not an option here.

The primary reason why Ricky might want to play more than 2 tiles is to ensure a good finish next move (i.e. going out). This also assumes that Tony will not bingo – i.e. it’s 50-50 on that, unless again Ricky blocks one or two bingo lines. Quackle’s top suggestion for this option is U(N)AI 12L which does block 2 possible bingo racks and gives him a chance at going out next move – however the chances of this happening (Tony not playing out with a bingo and Ricky drawing a good finisher) are not very high.

Quackle’s Take

Here’s the result of a many-ply simulation:


(click to enlarge)

I repeated the simulation once (with less plies) and the results are very similar (in terms of ranking).

(D)UI does simulate best with (S)UI following closely behind, and (B)I/I(N) simulates close to 42.60 (what I calculated earlier). Perhaps unexpectedly, a two-tile fish is the winning-est here.

One comment

  1. The actual game ending was:-Ricky played off 2 tiles at O1 ANI and emptied the bag.15H TOILETRY ended the game.

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