|
11
Nerd-Its
|
|
article
by Everyone has those nagging suspicions about repeating patterns of numbers or "rigging" the lottery. With eleven years of winning MegaMillions lottery numbers to analyze, this article examines some trends behind the jackpots based on such statistics as numerical distribution, number relationships and win frequency to name a few. Perhaps given some insight into the winning patterns, MegaMillions players just might have an improved chance of earning back their dollar with interest.
Add a Comment (45)
Email This
Message Author
Hi, first of all congratulations on your deep analysis.
I made something similar on the Spanish Lotto may years ago but using a different approach.
I started asumming the game was fair and the lottery wasn't rigged. Then I compared every winning combination (and sub-combinations of 3, 4 and 5 winning numbers) with the expected number of winners for the total bets on each game each week.
Using this method I confirmed a theory many people (myself included) has: that not all the players play their numbers "randomly" but that they have some "favourite numbers" and others that people just don't like. Combos like 1-2-3-4-5 are played a lot, also date combinations -- while numbers from 31 seem to be played less -- leading to too-many-winners or no-winner scenarios in each case.
You can detect this by checking how many winners are in each prize category on each game, comparing with the expected statistical results (not only for the big prize but for the smaller ones also).
On the Spanish Lotto (a basic 6/49 lotto, with 1 in 14 million odds) sometimes you have no big prize winners even when there are 30 or 40 million bets (and there "should be" at least 1 or 2); sometimes you may have 10 or 20 winners with only 10-15 million bets (for an «easy» combo like 1-2-4-8-9-24 or 7-14-21-28-41-42).
People can select their own numbers or let the «machine» select it at the shop, those numbers are supposed to be random and doesn't seem to influence on this (even if they are large).
If you read Spanish or can find a good translator you can find my (totally amateur) work here
http://www.microsiervos.com/archivo/azar/loto-un-sistema.html
-- Alvy
I have read the article and would like to ask if you think that the results can be used to predict any trends in future outcomes of the lottery?
If not, what test would be necessary to prove that the balls are not evenly distributed?
I have read the article and would like to ask if you think that the results can be used to predict any trends in future outcomes of the lottery?
I certainly hope so - I'd like to retire at 29. :-)
The data derived from this analysis was loaded into reference tables whereupon every conceivable lottery number was processed by my Mac Mini to assign a score. I'm still tweaking my scoring metric to balance out the results to something realistic. It takes about 36 hours from import for a complete process to transpire. I'll know in a few weeks if the use of these targeted numbers provides any more "luck" than my previous use of QuickPicks.
If not, what test would be necessary to prove that the balls are not evenly distributed?
I would think the examination for standard deviation from number of appearances would give a good indication for errant distributions. A comment on Slashdot was accurate in that while the data spans for years ... there really aren't that many numbers to work with! So it may just be the case that not enough drawings have transpired to either truly level out the distribution or to truly show the skew.
Any predictions?
I have a couple - but if I share those up front, I'll have to split the jackpot by the number of OmniNerd readers!
In the meantime, you can make some reasonable adjustments to haphazard picking by selecting numbers based on the presented graphs. The deeper analysis is relating the graphs to one another - but at a minimum it's better than rolling dice.
Assuming a completely fair game could a player still benefit from playing numbers that were played less frequently by other players? For example, you could play more numbers higher than 31 which are not dates, so there are fewer other players to share a prize with, assuming other players play 1 to 31 more frequently. Not only would the winning numbers be looked at but also the number of winners for a given combination or number.
Assuming a completely fair game could a player still benefit from playing numbers that were played less frequently by other players? For example, you could play more numbers higher than 31 which are not dates, so there are fewer other players to share a prize with, assuming other players play 1 to 31 more frequently.
Definitely so - in a completely fair game - playing less popular numbers increases your personal winnings by not having to share with everyone else. Unfortunately, the lottery website offers no data at all on the numbers that were actually played by everyone. THAT would be some cool analysis.
Absolutely. If { 1, 2, 3, 4, 5, 6 } ever hits, watch for the fireworks.
In the UK's National Lottery, that combination is played by over 10,000 people every week. The more people who play a combination, the more who have to split the big prize.
The chances of winning the big prize are pretty miniscule already - why choose numbers that are likely to cut your prize in half, or even more?
Absolutely. If { 1, 2, 3, 4, 5, 6 } ever hits, watch for the fireworks.
In the UK's National Lottery, that combination is played by over 10,000 people every week.
And that leads to an interesting question. It would be very interesting to actually be able to look at the pool of numbers people played and contrast it against each particular lottery. Even better than trying to see if numbers are distributed evenly over time is if they consistently skew away from trends selected by the masses. That sort of analysis would give the conspiracy theorists all the steam they need to believe the system was rigged.
Some of those numbers are published, though not all. The lottery advertisers actually do appear to make some effort to skew the picks.
The advertising largely pitches the idea that a player "owns" some numbers - they're "his" numbers. You see ads that say things like "don't let your numbers win without you", pushing players to play more often. That tends to get people "locked" into certain number patterns, that aren't chosen randomly.
Add to that the fact that people are really bad at picking a random set of numbers, even when they're trying, and the lotto ticket purchases are very skewed. There was an occasion in the UK when 113 people split the prize. The particular combination was pretty much what you'd expect a lot of people to do to choose "random" numbers from the form. Each number was from a different row and they were all from the second and third column - nearly alternating.
In March 2005, in New York, 110 people won the second-place prize in the Powerball game. It turned out that a fortune-cookie company had printed five of the six correct values on thousands of fortune cookies. Since the second-prize isn't a "parimutuel" prize (shared among the winners), it ended up amounting to around $19.5M in payoffs.
Turns out I did this exact same analysis (but i didnt think to write it up, nor that anyone would be interested in reading it).
At the time I did it (well over a year ago), 32 was the most common winning number and 3 was the most common mega ball.
The problem than, and the problem now, is that there havent been enough drawings to see that the histogram is truly flat or not (truly random).
I would suggest that, the drawing is in fact random, and choosing least common numbers will be better than choosing more common numbers. the reason for this is that eventually we would expect the histogram to be flat, in time the occurrence of infrequently drawn numbers should increase so as to flatten out the histogram.
However, each drawing is random. Before we see another 35 (the least common number when I did it), we could see 20 more 32s.
This was actually the intent behind the analysis at the bottom of the article. Rather than simple look at the distribution of all numbers based on their occurrence, each number was looked at individually for the frequency of occurrence based on time. So, using the data derived from that table, it can be determined which numbers are either winning a lot more frequently than they normally do or which numbers are "due" for a win. The query can be written in so many ways.
Each little piece of the analysis on its own would only let a player choose to play based on a style - distribution, pairs, frequency, etc. I'm still tinkering with the scoring metric to relate all the tables together. While I've already processed every possible lottery number and scored them against the reference tables, I still need to tweak that algorithm so that particular tables aren't weighted so heavily as to make the others irrelevant.
I'll know in a few weeks with additional passes of data whether or not the metrics actually help at all.
I would suggest that, the drawing is in fact random, and choosing least common numbers will be better than choosing more common numbers. the reason for this is that eventually we would expect the histogram to be flat, in time the occurrence of infrequently drawn numbers should increase so as to flatten out the histogram.
Bzzt! Incorrect. You lose all your accumulated points, and the Broyhill dinette set.
This is a surprisingly common fallacy - the idea that, in a fair game, the infrequent events will somehow "catch up", eventually. But, of course, it's completely wrong. For it to happen, it would mean that the machines somehow "remember" what the prior outcomes were, and change their behavior in response.
This is a surprisingly common fallacy - the idea that, in a fair game, the infrequent events will somehow "catch up", eventually. But, of course, it's completely wrong.
I've got to disagree with you there. For short periods of time, there will be a couple of events that will occur more often then the rest, and often dramatically so. Even for a uniform distribution. However, over a long time, the number of times that a particular event occurs is not much different then every other event, again assuming uniform probabalities. So, yes, in effect, the infrequent events do "catch up," eventually. But, the time scales for the infrequent events to catch up is related to the number of events possible, and increases quite dramatically with size.
That said, as you pointed out each event is independent, so picking the infrequently drawn numbers does not increase your chances of winning (the probability distribution does not change from event to event), even though the histogram will eventually flatten out.
For short periods of time, there will be a couple of events that will occur more often then the rest, and often dramatically so.
Yes, but that wasn't the conjecture.
If you imagine an infinite sequence of ball drawings, then the law of large numbers says that the sequence will contain an equal number of occurrences for each possible outcome. If you split that sequence into a finite prefix, and an infinite suffix, then the finite prefix will almost certainly not have an equal number of occurrences for each output - but the distribution in the suffix hasn't changed one bit.
The post to which I replied suggested that, having observed a fixed prefix, you've gained information about future drawings. It suggested that if 35 was overrepresented in the prefix and 22 was underrepresented, then in a similarly sized prefix taken from the remainder, 22 should be overrepresented and 35 underrepresented in some sort of "compensation" for the earlier distribution.
To simplify, he was saying that, if you flip a fair coin 9 times and it comes up heads 9 times, then the 10th flip is more likely to come up tails. In order to "catch up", the coin would have to have a memory of its prior behavior, and that memory would have to affect future behavior.
The infrequent events never "catch up" in a strictly numerical sense - what happens is that the proportion by which the distribution deviates from the theoretical expectation (the degree to which it's "out of whack") tends to zero as the sample size grows. If you flip that coin and it comes up heads 10 times, you could think of it as 10 tails "behind" - that's dramatic after 10 flips, but if you've flipped another 990 times and it came up 495 heads and 495 tails, you're still 10 tails "behind", but it's well within the expected outcome.
The reality is that there's no actual test by which you can prove a coin to be "fair" - so if you flip it 9 times and it comes up heads 9 times, you've received some pretty strong evidence that the coin is not fair and you should probably bet on it coming up heads yet again - but thinking that tails is somehow "due" has no logical support whatsoever.
The infrequent events never "catch up" in a strictly numerical sense - what happens is that the proportion by which the distribution deviates from the theoretical expectation (the degree to which it's "out of whack") tends to zero as the sample size grows. If you flip that coin and it comes up heads 10 times, you could think of it as 10 tails "behind" - that's dramatic after 10 flips, but if you've flipped another 990 times and it came up 495 heads and 495 tails, you're still 10 tails "behind", but it's well within the expected outcome.
Right. But, in a proportional sense, they do catch up, as 495/1000 is pretty close to 505/1000. Even in a strictly numerical sense tails will catch up. But, only over a very long time scale, and you cannot predict when it will occur, only its likelihood. (At some point, you expect for the number of tails to overcome the number of heads, and for them to "fight" for the most number of occurences.) I was trying to point out that A. Noni Moose was correct in stating that the histogram would eventually flatten out, which you seemed to disregard in your initial post. However, I wasn't disagreeing with the main thrust of your argument, that tails is not "due" in any finite time frame, but I was merely trying to clarify what exactly you were arguing against.
But, in a proportional sense, they do catch up, as 495/1000 is pretty close to 505/1000.
Yep.
Even in a strictly numerical sense tails will catch up. But, only over a very long time scale, and you cannot predict when it will occur, only its likelihood. (At some point, you expect for the number of tails to overcome the number of heads, and for them to "fight" for the most number of occurences.)
I don't think that really characterizes it properly. Yes - if you continued producing events (coin tosses) infinitely, then eventually tails will catch up - because if the list is infinite, then at some point every possible finite sequence will occur, and occur infinitely many times. So if tails are "ahead" by 10, then it's guaranteed that at some point in the future a finite run will have 10 more heads than tails, making it catch up. However, in any finite run, no matter how long, there's no guarantee the heads can ever catch up.
If you've flipped the coin 100 times, and tails is ahead by 20, you can compute the precise probability that heads will catch up within the next n trials, and it's never unity. In fact, if you look at the midpoint of any sequence - let's say a million flips - to see which one's ahead, then the probability they'll eventually become equal again in the second half of the finite sequence is only 0.5. That is, if we stop after a half-million flips to ask which one's ahead, there's only a 50:50 chance that it'll ever fall behind in the next half-million flips.
The non-intuitive bit here is that they really don't "fight" for the most number of occurrences - or at least, they don't fight very hard. According to Richard Epstein's The Theory of Gambling and Statistical Logic, the probability that the fraction of the time tails are ahead in a run of n tosses is less than k approaches (2/π) arcsin k½. For large n, with probability 0.5, the more fortunate side will be in the lead for over 85% of the run.
Unless you can talk about an infinite future, the "catch up" notion doesn't work.
I don't know a lot about statistics but I do know they are blind. The past cannot influence the future when it comes to randomness. A simpler analogy of this is if you roll a six sided die and come up with a five 7 times in a row; what are your chances of rolling another five? 1 in 6 again, because the die cannot 'remember' what it last rolled.
but if the die becomes deformed - it DOES remember what it did.
that is also taken partially into account here by the question of whether a drawn ball gets larger/smaller whatever.
even a flipped coin eventually suffers some wear and tear.
it is that wear and tear that will at some point skew the results away from truly random.
True - though, in order for this to be useful, the analyst has to know which machine was used to draw each ball. I think the organizations that actually run the lottery probably track that, but they definitely don't publish it.
I read the article and the first thing I noticed is that he didn't account for the number of times that there wasn't any mega millions winner...you know, how many weeks went by without a jackpot winner...where are those "no winner" numbers in his analysis?
Has anybody seen this - it's about using mixing errors in lotto picks:
http://use4.com/Prove-it.html
Kind of interesting, but not particularly useful. The article doesn't try to characterize the significance of the deviation they've found, and just on a gut feel, I think it would be very hard to exploit it in any significant way.
The article does make something of an error in talking about shuffling cards - the mathematical model that says a deck is randomized after seven shuffles assumes that the person doing the shuffle is reasonably skilled. The model assumes that the deck is cut approximately in half (the actual amounts in each half are modeled by one random distribution), then the model assumes that a small number of cards (typically 1-4), again expressed by a random distribution, is released from one half (specifically, the half that was originally on top), then another random small number from the other, which is repeated until both halves are depleted.
In practice, if the cards are old, the distribution of 1-4 cards tends to get skewed towards higher numbers. Many shufflers have a habit of not letting the cards fall first from the half of the deck that started on top, which means that the card previously on the bottom cannot move, and increases the probability that the card on top doesn't move, either.
If you ever watch a casino dealer shuffling for a game that only uses one deck - like poker - they almost never just do seven riffle shuffles. They do both riffle and overhand shuffles, and they also take the whole deck spread face down on the table and mix the cards around. And of course, more and more casinos are shuffling by machine, which are designed to approximate the mathematical model.
In what area did you include the "luck" factor? The statistical analysis appears complete; except... in all areas of gathered factual information there tends to be a human factor skewing the results in one fashion or another. Might the human factor in your analysis be "luck"? After all, it appears that some people are simply luckier than others.
Meh... it "appears" that the sun goes 'round the earth - but it ain't true. What you're talking about is reporting bias and ordinary random variation.
Some people are "luckier" than others in the sense that somebody has to win the lottery - isn't he lucky? But that's not something you can identify beforehand.
Others are "luckier" in the sense that things seem to fall in their favor - but it's likely someone's just ignoring the cases where they don't.
Does anyone have data showing the distribution of tickets vs winners? I remember, a while back, that it was skewed towards Georgia for some reason...
It would be interesting to know the stats on how many people play every week for their ENTIRE lives and die never having won any significant amount...
Some lottery corporations are changing the balls set over time.
Some lottery corporations are changing the balls set over time.
True - but you can still rig a lottery even with fresh balls. It's amazing how many organizations I've gone through where it only takes one bribe to completely crush the system of trust.
New balls, same guy weighing down particular numbers with a needled liquid or gas injection ... it could happen.
Please tell me when you start working for the State Lottery. I want in.
yes, powerball has 4 sets of white balls and 3 sets of red balls that they rotate in a certain fashion. i doubt its sequential which makes finding a pattern impossible. they have this info listed on their website under history if anyone is interested. they do give a date for the changing out of the sets of balls which could help in determining what data to use...
RSS
Where's the analysis? by Anonymous :: NR0 :: on 01 November 2007
You've collected and summarized a bunch of data here, but it would be much more useful (and straightforward) to run some inferential procedures (Monte Carlo procedures would be particularly easy to implement in this case) to see if these results were compatible with the hypothesis of a fair game. Give it a try!
It's in the MySQL database at home ... safe and sound. by VnutZ :: NR8 :: on 01 November 2007
You've collected and summarized a bunch of data here, but it would be much more useful (and straightforward) to run some inferential procedures (Monte Carlo procedures would be particularly easy to implement in this case) to see if these results were compatible with the hypothesis of a fair game. Give it a try!
Actually - using some optimized queries in MySQL against the tables that were created in the analysis allowed to rank score every conceivable lottery number. Optimization was needed to get the run time down from one year to about 36 hours. But, you don't think I'm going to give away MY number and share that -cough- winning -cough- with everybody?
RE: It's in the MySQL database at home ... safe and sound. by Anonymous :: NR0 :: on 01 November 2007
Are these numbers stored in the order they were drawn? I find it impossible to analyze data that is organized in ascending order. Please let me know. I have theories on pattern analysis. Oh yeah, don't forget to factor in leap year...
Number Storage by VnutZ :: NR8 :: on 01 November 2007
Are these numbers stored in the order they were drawn? I find it impossible to analyze data that is organized in ascending order. Please let me know.
You mean the master list of winning numbers? Yes, they are only available from the website in ascending order. But ultimately, that doesn't matter as the lottery is based on non-repeating combination rather than non-repeating permutation. You are right, however, in that there are other analysis tricks that could be studied if we knew the real drawing order ... but in terms of picking a possible winner it doesn't (well shouldn't) make a difference.
RE: Number Storage by Anonymous :: NR0 :: on 03 November 2007
They are available in order drawn on the website, not the data download.
RE: Where's the analysis? by Anonymous :: NR0 :: on 01 November 2007
I did do some analysis. Compensating for the four (ouch) different types, I get an overall p-value of 0.792. This means there is absolutely NO statistical significance...
You can test it yourself if you have R:
#Get big.dat at http://www.state.nj.us/lottery/data/big.dat big=read.table("big.dat",sep="%",fill=T) big$date=as.Date(apply(big[,1:3],1,paste,collapse="-")) big$type=ifelse(big$date>"1999-1-13",ifelse(big$date>"2002-3-15",ifelse(big$date>"2005-06-22",4,3),2),1) big$maxnorm=c(50,50,52,56)[big$type] big$maxspecial=c(25,35,52,46)[big$type] maxnorms=table(big$maxnorm) p=rep(0,56) for(i in 1:nrow(maxnorms)) p[1:as.numeric(names(maxnorms)[i])]=p[1:as.numeric(names(maxnorms)[i])]+maxnorms[i]*5 maxspecial=table(big$maxspecial) for(i in 1:nrow(maxspecial)) p[1:as.numeric(names(maxspecial)[i])]=p[1:as.numeric(names(maxspecial)[i])]+maxspecial[i] p=prop.table(p) allnum=unlist(big[,5:10]) t=table(allnum) chisq.test(t,p=p) plot(t/p)P
RE: Where's the analysis? by VnutZ :: NR8 :: on 01 November 2007
I don't suppose you could run that one last time - just restrict the data set to everything newer than June 25, 2005 (I think). That would be version 4, the current game, of MegaMillions.
I'll be the first to admit - statistics was the first math class I actually bombed ... only got a B- and that was years ago. So there are undoubtedly things I did in this analysis that would make a statistician cringe. I'm hoping more that the aggregation of all the different tables will result in something mildly more meaningful that straight random.
RE: Where's the analysis? by Anonymous :: NR0 :: on 01 November 2007
Hmm, there are of course two problems if actual balls are used. The normal balls can be unequal ('nope', p-value 0.5131 for v4). And the special balls can be unequal (0.8722).
Ok, the p-values are not significant, but... If actual balls are being used, there will be really tiny differences between balls. So one could currently select:
7 53 5 25 46 - 42
But if prices are shared between winners, one should not select those. Just select some random numbers in the middle...
On the other hand, it could also be that a ball that gets selected is being a little damaged, after which it becomes less likely to be selected. But I expect that a ball that is selected gets smaller and gets even more likely to be selected next time. I did not test for those type of effects yet. :)
Next to that, there could be interactions between balls that get selected together/not together more often. I did not test for them too. Replacement frequency and such is also of influence. Perhaps time for some physical tests? :)
Code (with a small fix):
big=read.table("big.dat",sep="%",fill=T) big$date=as.Date(apply(big[,1:3],1,paste,collapse="-")) big$type=ifelse(big$date>="1999-1-13",ifelse(big$date>="2002-3-15",ifelse(big$date>="2005-06-22",4,3),2),1) chisq.test(table(unlist(big[big$type==4,5:9])),p=rep(1/56,56)) chisq.test(table(unlist(big[big$type==4,10])),p=rep(1/46,46)) z=table(unlist(big[big$type==4,][,5:9])) z[order(z,decreasing=T)] z=table(unlist(big[big$type==4,][,10])) z[order(z,decreasing=T)]RE: Where's the analysis? by Anonymous :: NR0 :: on 02 November 2007
What I was looking for in my previous post was, the order and time the numbers were drawn. I wanted to use the analysis calculated by the earth's rotation including leap year minutes (x=(525,600*.25)+525,600) to see if there is indeed an prediction that can be achieved based on moments of time.
I know it is a bit much but I do not believe in randomnimity. I believe the word random was invented to fill in an explaination of things that happen on time intervals that we cannot explain. Thus my ultimate theory is that a number drawn can be predicted at the exact second, the trick is, finding the right calculation. Who knows if the Julian calendar is even the right approach?
I know it is out there and probably deemed cooky, but what if?
-IVX
RE: Where's the analysis? by VnutZ :: NR8 :: on 02 November 2007
Woah ... that's pretty deep. Might need the Monkey Computer to handle that one!
I think, however, that going off the deep end of prediction would find more immediate relevance in comparing weather conditions (barometric pressure specifically and rise or fall). Additionally, I think there is less value in the Earth's rotation while there would be more of an effect from the moon's position and the annual proximity to the sun. All way out there but it would of course, be interesting.
RE: Where's the analysis? by scottb :: NR7 :: on 02 November 2007
Moreover, the lottery is drawn pretty close to the same time every day. It's done live on TV, in a scheduled slot. I doubt there's more than a few minutes variation in time-of-day from drawing to drawing. How accurate might you have to measure to get an input with sufficient precision to make useful predictions? Milliseconds? Picoseconds?
If the relationship between time of day and the drawing is deterministic, but it's that non-linearly dependent on the conditions (the precise picosecond), I doubt you could prove it - even if you had access to the machines to instrument them and a huge experimental budget.
RE: Where's the analysis? by VnutZ :: NR8 :: on 02 November 2007
You forget ... a distributed network of monkey processing slaves. We can simulate anything with their collective powers so long they don't get distracted in a poop-fight.
The lottery does move around occasionally. Didn't they make some provision to make the $370 million drawing out of NYC? But, analyzing the location is equivalent to simply looking at the outcome of different machines - all things no public data exists for.
RE: Where's the analysis? by scottb :: NR7 :: on 02 November 2007
Didn't they make some provision to make the $370 million drawing out of NYC?
No idea. I don't actually follow the lottery itself (though you might not know it from how much I've commented on the topic) - I just find the math behind it interesting.
But, analyzing the location is equivalent to simply looking at the outcome of different machines
Approximately - though, if I understand it right, they typically have more machines than they use in any one drawing, and they use them in some sort of rotation schedule (though probably not just round-robin). One set of data I looked at (for the UK National Lottery) had actual names for the individual machines... "Galahad", "Lancelot", and so on.
Frankly, being able to associate a particular ball-drawing with a particular machine is going to be vastly more useful than knowing where and at what time the drawing occurred. It's far more reasonable to expect that manufacturing errors or wear and tear or even cheating could lead to better predictability than to think that any amount of precise measurement of the time or location of the drawing.
RE: Where's the analysis? by Occams :: NR6 :: on 23 March 2008
I can't get over the fact that every ball has an equal chance of coming out. Therefore the sequence 1,2,3,4,5,6 is just as likely to come up as any other sequence. But it never does, so that demonstrates how likely you are to choose a winning sequence.
No amount of ex-post pattern analysis can change that.
RE: Where's the analysis? by Anonymous :: NR0 :: on 12 April 2008
I may have missed this in your graphs and charts, (my understanding of math is dubious at best) but I was wondering if you noticed what I noticed. In 2001 I noodled around with the numbers for 2 years in an excel sheet to see what numbers came up the most, and after mapping out the numbers by 3 month seasons I saw an ebb and tide pattern to the grouping of numbers.
For example, the first 5 numbers would have low sequences (mostly from 1 to 20) for couple of weeks, then it would crest to high sequences (mostly 21 and up) for a month and a half or so, and it kept repeating this over the previous couple of years. The Megaball did the same thing, but in an reverse pattern, staying mostly low, and cycling up for a couple of weeks.
At that point I was stuck on how to come up with playable numbers within the two different groupings and never went further.
By the way good thread! And did it ever work out for you?
TMX
themysteriousx@sbcglobal.net