From dmorgan@ross.org Thu Dec 20 15:05:00 2001 Received: from mail.ross.org (mail.ross.org [205.232.30.10]) by mail.math.niu.edu (8.9.3/8.9.3) with ESMTP id PAA09175 for ; Thu, 20 Dec 2001 15:04:58 -0600 (CST) Message-id: Date: Thu, 20 Dec 2001 16:04:18 -0500 Subject: Chutes and ladders? To: rusin@math.niu.edu From: "David Morgan" MIME-Version: 1.0 Content-type: multipart/mixed; boundary="--=_--b847bc86.0584de33.000004fd" Content-Length: 46087 Status: RO This is a multi-part message in MIME format. ----=_--b847bc86.0584de33.000004fd Content-type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit I did a search for "Chutes and Ladders" and "analysis" and found your web page... it said you're looking for more info on games like this. Here is a litle something I threw together today after one too many games of C&L with my two-year-old. Thought you might find it of interest! D. Morgan ======================================================= Dr. David L. Morgan The Ross School dmorgan@ross.org http://www.ross.org (631) 907-5351 ======================================================= ----=_--b847bc86.0584de33.000004fd Content-Type: application/msword; name="Chutes and Ladders.doc" Content-Transfer-Encoding: base64 Content-Disposition: attachment; filename="Chutes and Ladders.doc" 0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAPgADAP7/CQAGAAAAAAAAAAAAAAABAAAA PAAAAAAAAAAAEAAAPgAAAAEAAAD+////AAAAADsAAAD///////////////////// //////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////// 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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAA== ----=_--b847bc86.0584de33.000004fd-- From rusin@math.niu.edu Thu Dec 20 19:38:58 2001 Received: from vesuvius.math.niu.edu (vesuvius.math.niu.edu [131.156.3.93]) by mail.math.niu.edu (8.9.3/8.9.3) with ESMTP id TAA10465; Thu, 20 Dec 2001 19:38:58 -0600 (CST) Received: (from rusin@localhost) by vesuvius.math.niu.edu (8.9.3/8.9.3) id TAA27985; Thu, 20 Dec 2001 19:38:59 -0600 (CST) Date: Thu, 20 Dec 2001 19:38:59 -0600 (CST) Message-Id: <200112210138.TAA27985@vesuvius.math.niu.edu> From: rusin@math.niu.edu To: dmorgan@ross.org Subject: Re: Chutes and ladders? Cc: rusin@math.niu.edu Content-Length: 4516 Status: RO > Thought you might find it of interest! Yes indeed! Thank you very much for sending this. Would you mind if I put a link to it on my website? I can actually use the chute-and-latter data you game me to compute exact probabilities; a simulation is not necessary. These take a while to grind out, but I have some estimates handy. We can construct a 100x100 "transition matrix" B whose i,j entry shows the probability of ending up in cell j when starting in cell i. Almost all entries are either 0 or 1/6 of course. The powers of B then show the probability of moving from cell i to cell j in multiple moves. Our position after 1 move is a vector with nonzero entries (each equal to 1/6) in positions 2,3,5,6,14,23, so if we multiply this by B^n we discover the probability of being in each of the other squares after n+1 moves. In particular, we can calculate the _expected_ position, and the probability of having finished, after 1, 2, 3, ... moves. Here are a few of these: moves expected position probability of finish 1, 8.833333333333333333, 0 17, 62.587667050363884984, .19460852232472954151 33, 81.668819046624220971, .57849340139413381474 49, 90.806555613248920600, .78840311374923682447 65, 95.387676089588083672, .89384091593780505453 81, 97.686000669131685226, .94674005886363157411 97, 98.839068235430158069, .97327952655548031018 113, 99.417561385758757036, .98659435808325637523 Thereafter (in fact, after 50 moves) we get an excellent fit to the data with a formula which states that the probability of finishing on the Nth move is (.0770680411)(.9578063749)^N. For smaller numbers of moves it seems better to go back to the actual data from multiplying out powers of B. Here are some finer divisions of the data: moves expected position probability of finish 1, 8.833333333333333333, 0 5, 30.814171810699588483, .00102880658436213992 9, 45.248834852728242658, .02988331856805365036 13, 54.637320760887811020, .09993389420202767886 17, 62.587667050363884984, .19460852232472954151 21, 68.997771503807061253, .30278661166989908787 25, 74.067575984236833612, .40765065965777475717 29, 78.206230217993574529, .49974101396643407160 31, 81.668819046624220971, .57849340139413381474 So the most common stopping time is around 18 moves. Let me zoom in a little more on the data set in this range: moves expected position probability of finish 13, 54.637320760887811020, .09993389420202767886 14, 56.703998858105806006, .12119299971560306609 15, 58.741107464472157859, .14372719681157051722 16, 60.714751450793112757, .16830679756790209324 17, 62.587667050363884984, .19460852232472954151 18, 64.343416806903044682, .22165888157150892250 19, 65.987595685188580413, .24874260704712490137 20, 67.535235944340794513, .27574123498572915246 21, 68.997771503807061250, .30278661166989908787 22, 70.379129307579945267, .32982089149455171799 23, 71.680642444458213725, .35652136924900981882 24, 72.907061819879920995, .38252727449042899574 25, 74.067575984236833606, .40765065965777475713 26, 75.172278574319596161, .43189051649872173125 27, 76.228381061825859315, .45531138495643418817 28, 77.239279576618381399, .47793789516117342620 29, 78.206230217993574529, .49974101396643407160 Interestingly, though the most common game length is 19 moves (2.7% of games), the probability that the game takes exactly N moves is nearly constant in this interval and NOT monotonic! The next most common game lengths are 18 moves and then _21_ (not 20) moves. In this way we can compute precisely the probability of ending after N moves (through N = 51, say), and so can compute the expected game length in two parts: add (game length)*(probability) for lengths <=51, then add the sum of the geometric series indicated earlier. These sums are respectively 21.50354838 and 14.50077518, giving an expected game length of about 36.00432357 moves. Approximately :-) I tested the predicted frequencies of game lengths against a 100000 game sample with your program. Results agree within experimental error. There are other questions to ask of course but I'll save those for another day. In particular, when two people play simultaneously, the game ends when the FIRST person reaches 100, right? This can affect the length of play. It certainly helps reduce the probability that the game drags on for hundreds of turns, as will happen about 1% of the time for a person playing alone. Again, thanks very much for taking the trouble to encode the game. dave From dmorgan@ross.org Fri Dec 21 10:10:54 2001 Received: from mail.ross.org (mail.ross.org [205.232.30.10]) by mail.math.niu.edu (8.9.3/8.9.3) with ESMTP id KAA14560 for ; Fri, 21 Dec 2001 10:10:52 -0600 (CST) Message-id: Date: Fri, 21 Dec 2001 09:58:17 -0500 Subject: Re: Chutes and ladders? To: rusin@math.niu.edu From: "David Morgan" References: <200112210138.TAA27985@vesuvius.math.niu.edu> In-Reply-To: <200112210138.TAA27985@vesuvius.math.niu.edu> MIME-Version: 1.0 Content-type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit Content-Length: 2620 Status: RO rusin@math.niu.edu writes: >Yes indeed! Thank you very much for sending this. Would you mind if I >put a link to it on my website? Of course. I just got the idea yesterday morning, and it was nagging me, so I wasted the whole day working on the program. It looks like it's contagious - it seems like you spent a good chunk of the day yourself working on your response! > > > These sums are >respectively 21.50354838 and 14.50077518, giving an expected game >length of about 36.00432357 moves. Approximately :-) > Bravo... my next step was to try to come up with a more subtle statistical treatment that made better predictions than my initial naive models... looks like you beat me to it. Although it seems to me that computing the probabilities in a 100x100 transition matrix might be just as computationally intensive as just "playing" 100,000 games. I wonder if there isn't a middle ground? > >There are other questions to ask of course but I'll save those for >another day. In particular, when two people play simultaneously, the >game ends when the FIRST person reaches 100, right? This can affect >the length of play. It certainly helps reduce the probability that the >game drags on for hundreds of turns, as will happen about 1% of the >time for a person playing alone. Yes - I thought of that too. It seemed like once the stats were worked out for a SINGLE game, that it would be a simple statistical calculation to compute the average length of a double game. But being trained as a theoretical physicist, statistics aren't my best area... (The experimentalists usually have to deal with that stuff!)... so I didn't take up that problem myself. What interests me more is whether there are "hot spots" on the board that get more hits than others. The next thing I plan on adding to the program are stats for each individual square. I've always wondered the same thing about Monopoly. If I mannage to generate any interesting graphs, I'll send them along. > > >Again, thanks very much for taking the trouble to encode the game. It was fun - I'm just glad someone else was interested. I sent my wife (a writer, not a scientist) a copy of what I had done all day, and she thinks I'm nuts! Are there any math education journals that publish this kind of stuff? Dave Morgan ======================================================= Dr. David L. Morgan The Ross School dmorgan@ross.org http://www.ross.org (631) 907-5351 ======================================================= From dmorgan@ross.org Fri Dec 21 11:06:00 2001 Received: from mail.ross.org (mail.ross.org [205.232.30.10]) by mail.math.niu.edu (8.9.3/8.9.3) with ESMTP id LAA14996 for ; Fri, 21 Dec 2001 11:05:59 -0600 (CST) Message-id: Date: Fri, 21 Dec 2001 12:05:58 -0500 Subject: Re: Chutes and ladders? To: rusin@math.niu.edu From: "David Morgan" References: <200112211632.KAA29731@vesuvius.math.niu.edu> In-Reply-To: <200112211632.KAA29731@vesuvius.math.niu.edu> MIME-Version: 1.0 Content-type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit Content-Length: 1056 Status: RO rusin@math.niu.edu writes: > >I looked at the problem again later in the day and realized I could >compute some of these numbers exactly with a little more algebra. >The average game length really is 36.0something, but can be >expressed as an exact fraction. I've put a summary of the information >on the same web page, >http://www.math.niu.edu/~rusin/uses-math/games/other/ I'll definitely take a look at the calculations... in the meantime... I added those square-by square stats to my program, and there are definitely some interesting "hot spots" on the board. ( http://mail.ross.org/~dmorgan/chutes/squarestats.jpg ) I've yet to take a closer look to see how they correlate with the endpoints of the chutes and ladders. > D.M. ======================================================= Dr. David L. Morgan The Ross School dmorgan@ross.org http://www.ross.org (631) 907-5351 =======================================================