Expert Heads Up No Limit Hold’em v.2: Strategies for Multiple Streets by Will Tipton - Page 4 - Books and Publications

irishman, a half pot check raise risks much more than a half pot bet. Half pot raise implies calling whatever is bet and then raising 1/2 of the new pot. So say pot is 10 and your opponent bets 7, a half pot check raise is a raise to 19 (7+.5*(10+7+7)). 19 to win 17 so naive defend frequency is 47.2%. Naive defend frequency for a half pot raise will vary between 40% and 60% for 2x pot to 1/10th pot betsizings.

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06-01-2014 , 11:49 AM

#77

Irishman07

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Join Date: Jan 2007 Posts: 1,001

Quote:

Originally Posted by stevepa

oh yeah that was dumb of me. thanks

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06-03-2014 , 10:04 AM

#78

Mashed

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Join Date: Jun 2014 Posts: 18

Hello guys,

Is necesary to read the Vol 1 for understand the Vol 2? Or this is just a update?

Thank you!

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06-03-2014 , 11:02 AM

#79

King of the North

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Join Date: May 2014 Posts: 149

Two separate books. Buy the first, first.

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06-03-2014 , 11:35 AM

#80

Mashed

newbie

Join Date: Jun 2014 Posts: 18

TY King!

I will buy first de vol 1!

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06-03-2014 , 11:57 AM

#81

mmowgli

newbie

Join Date: Nov 2013 Posts: 21

Quote:

Originally Posted by pasita

Anyone care to help me here, can't see what I'm missing? If BB always checks everything, shouldn't SB play like MoP 11.3 (the half street game), i.e. bet only top 22% and bottom 11%? If BB sometimes plays that and sometimes has a bet out strategy, how can SB switch directly to betting top 33% and bottom 17% ?

Hello pasita! I think it would help you if you look at figure 13.10 on page 240 in chapter 13.10 where the equilibrium for the river symmetric distributions case with B=S=P is represented.
If the BB always checks everything, your exploitative adjustment for the BB I think it's right (that situation is described in Vol.1) and if the BB only checks sometimes (like it happens at equilibrium), betting top 33% and bottom 17% is the equilibrium response facing a check.
But what you quote from the book I think it's just explaining that the best 1/3 of the BB range is indifferent between checking and betting ONLY if the SB is playing at equilibrium and it says to us that with that part of our range we must play a mixed strategy at equilibrium, BUT doing it with the wrong ratio bet:check would make us exploitable.

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06-03-2014 , 12:38 PM

#82

pasita

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Join Date: Mar 2007 Posts: 1,101

Sorry, still does not compute. Which might be a problem at my end, don't get me wrong.
F13.10 describes the full street 1 psb symmetric game, same solution as in MoP. BB ALWAYS bets his top and bottom, x/c some and x/f some. This is an equilibrium to my understanding. I still don't see
a) why BB should sometimes (how often?) check (a part of ?) his top hands if he's at equilibrium already using the strategy from f13.10
or
b) why SB would always give action with 50% of his range if BB sometimes (100%? 99%? 50%? 1%?) plays the "check everything game", which, at least IMO, equals to the half street game (MoP 11.3).

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06-03-2014 , 03:28 PM

#83

mmowgli

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Join Date: Nov 2013 Posts: 21

a) In the figure 13.10 the BB is value-betting 1/6 of the time with his top hands and check-calling 2/6 of the time (50% minus 1/6 of the time). So between his 5/6th percentile and his 6/6th percentile he bets, and between his 4/6th percentile and his 5/6 percentile he checks. With the hands between 4/6th percentile and 6/6th percentile he is really indifferent at equilibrium because you are going to end all-in anyway vs the same number of hands better or worse than yours anyway.
b) I don't know why you think that the BB plays "check everything game", IMO this idea isn't in the book. The most profitable way to play the 0th percentile hand to the 1/12th percentile hand still would be betting as a bluff. And of course any variation in the structure of the BB ranges could have an exploitative response as you say if BB checks every hand, it's just the BB's top 1/3 hands would be always indiferent if SB doesn't change his strategy.

Last edited by mmowgli; 06-03-2014 at 03:34 PM.

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06-03-2014 , 05:18 PM

#84

pasita

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Join Date: Mar 2007 Posts: 1,101

This from the Book (p.45):

Quote:

Why is it that Hero's GTO strategy cannot involve either always checking or always leading here with all his near-nut hands?

This from above:

Quote:

This applies to a full 1/3 of the BB's range in this game (in the symmetric distributions case, with B=S=P). The top 1/3 of BB's range is actually indifferent between leading and check-calling at equilibrium, and indieed, he sometimes leads and sometimes checks with hands in this region.

I thought those imply that BB sometimes plays the "check everything": If BB sometimes checks those top hands, how can he still be betting the bottom 1/12th every time? (Not that those particular hands care, they're 0 EV anyway vs Nemesis.)
My thinking is that if BB sometimes checks the top 1/6 of his range (remember, he can tell SB this, and even the percentages, if this still is equilibrium), SB will now be value betting (and thus bluffing) tighter (converging on the half street game when BB always checking the top), and the top 1/6 of BB's range does get less action this way.

Anyway, my original question still remains: if the f13.10 indeed shows the equilibrium (as I believe), why does BB _have_ to still check the top of his range sometimes?

If Will would give us the strategy of how BB should check the top hands (always some? sometimes all? something different?), I could give a shot at the EVs. I'm saying "BB can always play the strategy of f13.10, no need to check those top hands". I'll also add "the game value for BB will be strictly worse if he checks some of the top 1/6 hands and announces his strategy to SB". Extremely happy if someone shows the math to prove those wrong.

BTW if anyone thinks this discussion should be moved to Theory to keep the book thread clean, I'm very happy with that too.

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06-03-2014 , 06:25 PM

#85

mmowgli

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Join Date: Nov 2013 Posts: 21

I think that when it talks about nuts or near-nuts it means those hands that always go all-in if SB wants it (the top 1/3 of the SB distribution that at equilibrium are playing the top 1/6 by betting and the next 1/6 by checking.)

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Why is it that Hero's GTO strategy cannot involve either always checking or always leading here with all his near-nut hands?

I think if the BB checks more hands than he does in the F10.13 the SB will bet less often facing a check like you say, and everything different to the figure 10.13 I suppose is not equilibrium and it could be exploited.

Quote:

If BB sometimes checks those top hands, how can he still be betting the bottom 1/12th every time?

Because if SB doens't change his GTO strategy (probably to call more often when facing a bet and bet less facing a check) it would be the most profitable way to play them. But this wouldn't be an equilibrium strategy for the BB!

Quote:

if the f13.10 indeed shows the equilibrium (as I believe), why does BB _have_ to still check the top of his range sometimes?

He never has to check the top 1/6. BB has to check to play equilibrium only what it is in the image. By top of his range, I suppose you mean nuts and near-nuts that in this situation is 1/3 top and in fact this is played the half top (1/6) by betting and the next half (2/6 top hand to 1/6 top hand) by checking.
I think this is the only equilibrium strategy for the BB, he is indifferent with that top 1/3 if SB continues with his strategy but if he changes something the F10.13 it would be exploitable.

Quote:

"the game value for BB will be strictly worse if he checks some of the top 1/6 hands and announces his strategy to SB"

Totally agree with that!
I also had a hard time understanding it a few days ago that is why I was very happy to try to help.

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06-03-2014 , 08:04 PM

#86

pasita

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Join Date: Mar 2007 Posts: 1,101

Quote:

Originally Posted by mmowgli

Quote:

He never has to check the top 1/6. BB has to check to play equilibrium only what it is in the image.

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I think this is the only equilibrium strategy for the BB

I agree with all of those. But the book implies otherwise, and the author says otherwise in this thread. (That, or I have serious reading comprehension problems.) He says the equilibrium isn't unique, but I haven't seen his videos where he discusses this, nor any math anywhere supporting this. Even if it wasn't unique, I don't get why BB would _need_ to deviate.... it's still an equilibrium.

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By top of his range, I suppose you mean nuts and near-nuts that in this situation is 1/3 top and in fact this is played the half top (1/6) by betting and the next half (2/6 top hand to 1/6 top hand) by checking.

I meant the top 1/6 which bets in the equilibrium of f13.10

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06-03-2014 , 08:30 PM

#87

pasita

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Join Date: Mar 2007 Posts: 1,101

Now wait a second. Does the book imply that you can bet ANY half of the top 2/6 range, i.e. 1/6 of total range, and still be at equilibrium? If that's what we're looking at, there might be infinite amount of equilibria there, yes. Times another infinite amount for choosing the bluffing range, as long as the frequencies stays right. It's just that those equilibria are as sensible as the GTO play of folding the nuts on the river at some branches.

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06-04-2014 , 11:29 AM

#88

mmowgli

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Join Date: Nov 2013 Posts: 21

Quote:

you can bet ANY half of the top 2/6 range, i.e. 1/6 of total range, and still be at equilibrium? If that's what we're looking at, there might be infinite amount of equilibria there, yes. Times another infinite amount for choosing the bluffing range, as long as the frequencies stays right.

I also think like that! IMO if the bluffing hands are picked up from the 1/6 bottom range with a frequency of 1/12 of the time and the value betting hands are taken from the 1/3 top range with a frequency of 1/6 of the time.

Last edited by mmowgli; 06-04-2014 at 11:39 AM.

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06-04-2014 , 09:43 PM

#89

yukoncpa

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Do we get the ebook free when we order the paperback version?

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06-04-2014 , 11:27 PM

#90

yaqh

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Quote:

Originally Posted by pasita

p81, BB holds the perfect mix of nuts and air.
"the SB's calling frequency would be completely unconstrained - any frequency between 0 and 100 percent would be co-optimal".
I'd think it's worth noting that if SB announces his strategy as "call 0%", BB can't exploit that, as he just doesn't have the needed extra bluffs to do that. But if SB announces his strategy as "call 100%", BB can now stop bluffing.
Shouldn't the "100%" be 1-alpha?

Quote:

Originally Posted by MrPete

My thoughts exactly! While BB can't decide to add any more bluffs given the range he gets there with, he certainly can decide to never bluff if SB calls more than 1-alpha.

You are correct -- looks like that should be any calling frequency from 0 up to P/(S+P).

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06-04-2014 , 11:33 PM

#91

yaqh

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Quote:

Originally Posted by Qlka

Will, could you share full solutions for pre-flop play in the same way you did for examples in volume 1? Textual representation of ranges where we could import it into equity visualiser would be helpful and knowing all decision points for sb and bb also.

Quote:

Originally Posted by yaqh

I believe the game trees I used (or at least the preflop part of them) are fully specified at the beginning of section 15.2 (pgs 335-7). But as far as the ranges in numerical format, yea I can do that -- give me a few days.

So, I looked into this, and it's actually a lot more inconvenient than I expected, given the size of those datasets. Sorry to go back on what I said, but I'm not going to be able to release that info in the near future. Sorry for the inconvenience.

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06-04-2014 , 11:58 PM

#92

yaqh

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Quote:

Originally Posted by Irishman07

pg. 267 of PDF. SB folds vs BB's half-pot check-raise more than the naive frequency of 33% and I'm not sure I understand why. If BB is deciding whether to bluff check-raise or fold and SB folds more than naive frequency, BB has incentive to never fold and always bluff check-raise, right?

I understand that on K73r BB's bluffs have less equity vs SB's continuing range than on QT9, so it seems to be that SB can fold relatively more on K73, but folding 42.3% of the time seems way too high. What am I missing?

Great book, btw. I'm really enjoying it so far.

Quote:

Originally Posted by stevepa

irishman, a half pot check raise risks much more than a half pot bet. Half pot raise implies calling whatever is bet and then raising 1/2 of the new pot. So say pot is 10 and your opponent bets 7, a half pot check raise is a raise to 19 (7+.5*(10+7+7)). 19 to win 17 so naive defend frequency is 47.2%. Naive defend frequency for a half pot raise will vary between 40% and 60% for 2x pot to 1/10th pot betsizings.

Yup, thanks stevepa! I really like the way of looking at it you mention at the end. I get, in general, a naive folding freq of (4B+P)/(6B+3P) for bluffing over a halfpot raise.

So in the situation in the book (a halfpot raise over a halfpot bet), the naive result is a folding freq of 1/2, and the actual folding freqs are all somewhat less, accounting for the fact that weak hands can still capture some of the pot after bluff-raising.

Glad to hear you're enjoying the book, Irishman.

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06-05-2014 , 12:01 AM

#93

yaqh

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Join Date: Aug 2007 Posts: 5,745

Quote:

Originally Posted by Mashed

Hello guys,

Is necesary to read the Vol 1 for understand the Vol 2? Or this is just a update?

Thank you!

Quote:

Originally Posted by King of the North

Two separate books. Buy the first, first.

Yup, I think it's fair to think of both together as one book -- vol1 goes from chapter 1-8, and vol2 goes from 9-17. I do start vol2 with a brief summary of the most important points from the first book, but it's probably best to read the first first.

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06-05-2014 , 01:00 AM

#94

yaqh

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Quote:

Originally Posted by yaqh

On pg 44, I said what I mean by "near-nut" hands -- a hand strong enough that we'll always try to get all-in, and whenever Villain has a better hand, he'll always try to get all-in too. Basically, it just means that whenever we have such a hand and Villain has better, we're guaranteed to go broke.

This applies to a full 1/3 of the BB's range in this game (in the symmetric distributions case, with B=S=P). The top 1/3 of BB's range is actually indifferent between leading and check-calling at equilibrium, and indieed, he sometimes leads and sometimes checks with hands in this region. And SB puts in the same amount of money, exactly, versus both actions.

In the way the solution is normally structured, BB leads with his strongest hands and check-calls some weaker ones, but that solution isn't actually unique. Other structures (which can involve check-calling with the nuts) are co-optimal. I talk about this and this game more in one of my videos.

Quote:

Originally Posted by pasita

Yep, I don't see how it applies. Can SB somehow exploit if BB always uses the structuring from MoP or A13.10?

No, co-optimal means multiple strategies are unexploitable.

Quote:

The co-optimality of the "check everything" from p44 and using the "bet-chcall-fold" regions came as a surprise to me.

I definitely don't say that checking his entire range is an unexploitable strategy of the BB's here. To be clear, it isn't.

If you don't see why, then let me know, but otherwise, I'll skip most of the text where you are assuming it is and confused because of it.

Quote:

Originally Posted by mmowgli

the best 1/3 of the BB range is indifferent between checking and betting ONLY if the SB is playing at equilibrium and it says to us that with that part of our range we must play a mixed strategy at equilibrium, BUT doing it with the wrong ratio bet:check would make us exploitable.

True if we remove the "ONLY".

This indifference is shown graphically in Figure 13.6.

Quote:

Originally Posted by pasita

The equilibrium strategy in Figure 13.10 does involve checking some of his top hands. There are other, co-optimal, strategies that also also involve checking and betting with some of his strongest hands, but not the exact same ones as in the strategy in the figure. These have the same EV at equilibrium.

Quote:

Originally Posted by mmowgli

yes

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b) I don't know why you think that the BB plays "check everything game", IMO this idea isn't in the book. The most profitable way to play the 0th percentile hand to the 1/12th percentile hand still would be betting as a bluff. And of course any variation in the structure of the BB ranges could have an exploitative response as you say if BB checks every hand, it's just the BB's top 1/3 hands would be always indiferent if SB doesn't change his strategy.

Actually, BB is (at least) indifferent to checking with each individual hand in his entire range at equilibrium (see Figure 13.6). His weakest 1/12 of hands, in particular, are indifferent between bluffing and check-folding. But that doesn't mean that going ahead and checking his entire range is unexploitable.

Quote:

Originally Posted by pasita

This from the Book (p.45):

Quote:

Why is it that Hero's GTO strategy cannot involve either always checking or always leading here with all his near-nut hands?

This from above:

Quote:

I thought those imply that BB sometimes plays the "check everything":

Tbh, it seems like that first quote explicitly says that he can't always check his near nuts, much less his whole range.

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If BB sometimes checks those top hands, how can he still be betting the bottom 1/12th every time?

He takes some of the good hands that used to check-call and bets with them instead. His overall value-leading frequency does not change.

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If Will would give us the strategy of how BB should check the top hands (always some? sometimes all? something different?), I could give a shot at the EVs. I'm saying "BB can always play the strategy of f13.10, no need to check those top hands". I'll also add "the game value for BB will be strictly worse if he checks some of the top 1/6 hands and announces his strategy to SB". Extremely happy if someone shows the math to prove those wrong.

Take the region of value-leading hands (which total 1/6 of his range) and move it to the left a bit. The strongest hands (which used to be leads) are now check-calls. Calculate EVs to verify that you're still at equilibrium.

And again, there's no need to deviate from the strategy in Figure 13.10 -- the other co-optimal strategies have the same EV at equilibrium.

Quote:

BTW if anyone thinks this discussion should be moved to Theory to keep the book thread clean, I'm very happy with that too.

This thread seems fine to me.

Quote:

Originally Posted by pasita

The equilibrium isn't unique, nowhere have I said that BB _needs_ to deviate, and the 65 minute video #6 in my (first and only) video pack is almost entirely dedicated to this model game.

Quote:

Originally Posted by pasita

There are an infinite number of co-optimal strategies.

A strategy involving folding the nuts on the river is dominated, i.e., there's some other strategy that's better (or at least as good as and sometimes better) than it, regardless of Villain's strategy. Those strategies can be equilibrium only if we never actually get to those river spots during equilibrium play.

I don't see how we can get any dominated equilibria by changing which of Hero's near-nuts check-call and which valuebet.

Last edited by yaqh; 06-05-2014 at 01:06 AM.

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06-05-2014 , 01:07 AM

#95

yaqh

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Quote:

Originally Posted by yukoncpa

Do we get the ebook free when we order the paperback version?

No, sorry.

(OK, I think I'm finally caught up with this thread now! If I missed anything up to this point, let me know.)

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06-05-2014 , 05:07 AM

#96

pasita

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Join Date: Mar 2007 Posts: 1,101

Ok, thanks for the clarification. Can't believe I fell for the infinite equilibria trap again. An earlier mention of "BB's betting frequency still always stays the same" somewhere would have saved me some headache

Quote:

I don't see how we can get any dominated equilibria by changing which of Hero's near-nuts check-call and which valuebet.

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Take the region of value-leading hands (which total 1/6 of his range) and move it to the left a bit.

It's just that if we do stuff like that, while still equilibrium, it tends to work worse against carbon based opponents than using the "sensible" equilibrium (as per f13.10). Pretty much like the "I'll fold the nuts as a GTO surprise, since this branch wasn't NE anyway and I just don't care"

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06-05-2014 , 07:30 AM

#97

mmowgli

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Join Date: Nov 2013 Posts: 21

Quote:

Originally Posted by mmowgli

But preflop when a lot of hands are indifferent at equilibrium, it means that each hand combination of the same hand probably should take an action with a non-zero probability (e.g JdTd and JcTc open-shove 31.4% of the time and limp 68.6% of the time, each one). I don't know any EQ visualizer that can do this.

Quote:

Originally Posted by yaqh

EDVis can if you input ranges using the Edit box in the (undocumented) numerical format

Might you show us the numerical format to input-for example a 20% of AKs? Really appreciate your help Will

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06-05-2014 , 01:33 PM

#98

minotaurs

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Join Date: May 2012 Posts: 193

Quote:

Originally Posted by yaqh

No, co-optimal means multiple strategies are unexploitable.

The equilibrium isn't unique, nowhere have I said that BB _needs_ to deviate, and the 65 minute video #6 in my (first and only) video pack is almost entirely dedicated to this model game.

First and only doesnt mean that there will be no video pack 2 right? Its just the only 1 out there for now right.

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06-06-2014 , 12:23 AM

#99

yaqh

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Quote:

Why?

At equilibrium, all those strong hands are indifferent between leading and check-calling, since the same amount of money goes in with both lines, on average. (That's a main point of the section.)

Against a human opponent, in any particular spot, one or the other is probably better -- Villain's going to put more money in vs either a bet or a check, and we should play all our strong hands whichever way that is.

But if we are going to play some strong hands each way (as we must at equilibrium), I can't see any reason why the structure in Fig 13.10 is more "sensible" than any other.

Quote

06-06-2014 , 12:25 AM

#100

yaqh

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Join Date: Aug 2007 Posts: 5,745

Quote:

Originally Posted by mmowgli

Might you show us the numerical format to input-for example a 20% of AKs? Really appreciate your help Will

Well, sure, here's some info on the numerical input format for EDVis in case anyone finds it helpful, but as I mentioned earlier, this is an undocumented feature and not one I plan on supporting, generally.

Spoiler:

Ok so, the syntax is a N: followed by 1326 colon-separated numbers. Each number is the fraction of one of the hand combos in the range. Here's a bash script that prints out the hands in the required order:

for rankA in `seq 2 14`
do
for rankB in `seq $rankA 14`
do
for suitA in `seq 1 4`
do
for suitB in `seq 1 4`
do
if [ $rankA == $rankB ]
then
if [ $suitA -ge $suitB ]
then
continue
fi
fi

nameA=$rankA
if [ $rankA -eq 10 ]; then nameA=T; fi
if [ $rankA -eq 11 ]; then nameA=J; fi
if [ $rankA -eq 12 ]; then nameA=Q; fi
if [ $rankA -eq 13 ]; then nameA=K; fi
if [ $rankA -eq 14 ]; then nameA=A; fi
nameB=$rankB
if [ $rankB -eq 10 ]; then nameB=T; fi
if [ $rankB -eq 11 ]; then nameB=J; fi
if [ $rankB -eq 12 ]; then nameB=Q; fi
if [ $rankB -eq 13 ]; then nameB=K; fi
if [ $rankB -eq 14 ]; then nameB=A; fi

if [ $suitA -eq 1 ]; then suitnameA=c; fi
if [ $suitA -eq 2 ]; then suitnameA=s; fi
if [ $suitA -eq 3 ]; then suitnameA=d; fi
if [ $suitA -eq 4 ]; then suitnameA=h; fi
if [ $suitB -eq 1 ]; then suitnameB=c; fi
if [ $suitB -eq 2 ]; then suitnameB=s; fi
if [ $suitB -eq 3 ]; then suitnameB=d; fi
if [ $suitB -eq 4 ]; then suitnameB=h; fi

echo $nameB$suitnameB $nameA$suitnameA
done
done
done
done

For example, here's 22+:
N:1:1:1:1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:1: 1:1:1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:1:1:1:1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:1:1:1: 1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:1:1:1: 1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:1:1:1:1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:1: 1:1:1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:1:1:1:1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:1:1:1:1:1:1:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:1:1:1:1:1:1:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:1:1:1:1:1:1:0:0:0:0:0:0:0:0: 0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:1: 1:1:1:1:1:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:0:1:1:1:1: 1:1

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