Dau: user manual
Contents
1. 22 REFERENCES 6 Minimizing the functions set Question whether it s easier to play Landau s game with more or with less functions allowed is open for discussion I chose to implement three levels of difficulty in DAU in form of three sets of functions allowed 6 1 Easy level of the game With all functions and operations listed in 4 1 all possible license plates are solvable as we ve seen general solutions from 5 1 and 5 3 are applicable Still some of the functions are almost never used and some seem redundant in the hard level they are disabled 6 2 Hard level of the game With functions listed in 4 2 not all license plates are solvable Double factorial is added to reduce number of unsolvable plates but I assume there are still more than a few in the game they make the game harder since you don t know where you must concentrate and find the answer and where trying is futile 6 3 The Teorminimum This level is named after the famous Landau s exam but has no other connec tion with it It forces the player to use logarithms intensively in order to do everything multiplication division exponentiation powers roots making re ciprocals everything can be done using logarithms here lies the answer to questions where s the log x button Y x button etc in all levels manipulate logarithms Thanks to the fact that trigonometry is allowed in this level all plates are solvable using general solution2. e Chat in game chatting in DAU can either be disabled default or enabled 15 3 2 Playing a multiplayer game 3 INTERFACE Game over Game over in 3 rounds OK Figure 16 End of game Set options Game options Number of rounds Unlimited Time per round sec Unlimited Level of difficulty Easy Your character Landau gt Chat Disabled Rules Faster player wins Connection options Your port 31416 Opponent s IP Opponent s port 31416 Cancel Offer rules Figure 17 Multiplayer options 16 3 2 Playing a multiplayer game 3 INTERFACE Game options Number of rounds 10 127 0 0 1 offers you the following rules Number of rounds 10 Time per round sec 30 Level of difficulty Teorminimum Yo ch Your character Landau Chat Enabled Ru Rules No need to hurry co Do you want to accept them Yo no essa Op y Opponent s port 31416 Cancel Offer rules Figure 18 Rules offer e Rules two types of game rules are possible Faster player wins default and No need to hurry In the first scenario only one player can take the point in one round and as soon as one answers correctly that round ends or when time expires or after both players give up the round then none of the players gets the point In the second one round doesn t end before both players give it up answer correctly or the time expires so both can get the point in
3. About button informs you about the author and technology used while Quit game button takes you back to the real world As far as Single player game and Multiplayer game buttons are concerned explanation follows 11 3 1 Playing a single player game 3 INTERFACE 3 1 Playing a single player game Choosing single player game in the main menu takes you to the options screen shown in figure 11 Four options are available for single player game rest of them are disabled since they are related to the multiplayer game Number of rounds how many license plates do you want to solve in the game options are 1 5 10 20 or unlimited default Unlimited means the game ends when you click the Quit game button there is no limit on the number of rounds Time per round sec how much time will you have for each license plate options are 10 20 30 60 or unlimited default Unlimited means the round ends if you skip it clicking on an appropriate button adds nothing to your score or give a correct answer adds one point to your score If 10 20 30 or 60 is chosen round also may end if the time runs out adds nothing to your score Level of difficulty how hard do you want the game to be options are Easy default Hard or Teorminimum Levels are described in detail in section 4 For now we ll just say that easy level gives you the biggest set of functions and operations for use on the digits while that set is reduced in other two l
4. 19 20 21 21 21 22 Part I The History Figure 2 Lifshitz left and Landau right on a car trip in the Caucasus region 1 CHARACTERS INVOLVED Figure 3 Lev Davidovich Landau 1 Characters involved Three curious characters appear in DAU and it would be fair to introduce them here Each one will be presented with a photo brief biography and an anecdote Of course it was hard to select only one anecdote per person especially in case of Lev Landau 1 1 Lev Davidovich Landau Lev Davidovich Landau Baku 1908 Moscow 1968 Soviet physicist Academi cian Hero of Socialist Labour Stalin Lenin and Nobel prize laureate Friends called him simply Dau He won the Nobel prize in 1962 for his work on super fluids He is well known for his work in quantum mechanics theory of phase transitions quantum electrodynamics Erudite and active thinker in every meaning of that word His school of theoretical physics was a great place for making of brilliant scientists his comprehensive exams in all branches of physics called The theoretical minimum shorter Teorminimum Only 43 men ever passed all the exams as you can see in the list in figure 4 Landau once attended a session of the Russian Academy of Sciences at which the notorious agronomist Trofim Lysenko the founder of creative Darwinism gave a lecture on the so called inheritance of acquired traits When the talk was over Landau asked a penetrating quest
5. bunch of anti communist bandits surrounded the town The leader was suspicious of Tamm who was dressed in city clothes He demanded to know what Tamm did for a living He explained that he was a university professor looking for food What subject the bandit leader asked Tamm replied I teach mathematics Mathematics said the leader OK Then give me an estimate of the error one makes by cutting off a Maclaurin series expansion at the nth term Do this and you will go free Fail and I will shoot you Tamm was not just a little astonished At gunpoint he managed to work out the answer He showed it to the bandit leader who perused it and then declared Correct Go home Tamm never discovered the name of the bandit 2 ORIGIN OF THE GAME Figure 8 Landau Lifshitz and Tamm 2 Origin of the game There is not much to say about the game origins Both Kaganov and Gorobets in 2 and 1 tell roughly the same story 2 1 The game is born Kaganov writes in 2 about the game Landau invented it played it and told others about it encouraging them to play Rules are simple in those days Soviet license plates looked like one in figure 1 four digits were combined in two pairs separated by a hyphen Landau applied elementary mathematical operations and functions on both sides using only digits given in order they re given so the hyphen could be replaced with a sign of equality For instance if the
6. en Round 1 started ma Score 0 Landau Lifshitz Ti Round ime 7 oun 1 Resign round Ask Tamm L Quit match 7 WEN sin cos tan exp EE S5 asin acos atan In EE 2 3 505 Lg lbs sart ete Send O CE Leftsideediting O4 Figure 12 Game screen easy level 13 3 1 Playing a single player game 3 INTERFACE Round 1 started en Score aly ch Landau Lifshitz Time 4 Round 1 7 Resign round Ask Tamm OK Quit match 7 3 9 a E sin o ow lt C C Mi MN sme hen 3 S Os Be 5 sa in CE Leftsideediting 53 04 Figure 13 Game screen hard level e Round 1 started Score 0 Landau Lifshitz Time Round 4 1 m Resign round SK Ta mm Help OK Quit match N 7 8 sin cos tan asin acos atan aan NE NUN CE Leftsideediting 04 Figure 14 Game screen Teorminimum level 14 3 2 Playing a multiplayer game 3 INTERFACE Round 1 started en Score 0 Landau Lifshitz OK sin cos tan exp bas asin acos atan In Left side editing A amp lg Ib sqrt 53 04 Ib Ib 3 6 lb 243 OK Figure 15 Tamm answers Under the in game menu is chat box disabled in single player game On the right we hav
7. license plates were 12 34 potential solutions would be 1 2 3 4 or 1x 2 3 4 or 1 2 3 v4 Kaganov wondered if this game always had a solution Landau s answer was negative not because he proved that general solution doesn t exist but because he didn t solve all the license plates he saw So do general solutions exist Answer is in the part III of the manual 2 2 What can be used Landau made it a rule functions and operations used in the game should be elementary and known to high school students It s fairly obvious that differ entiation would trivialize the game reducing every combination to 0 0 but since that s not elementary it s not allowed Similarly integer part of a number denoted by and fractional part denoted by also trivialize the game so 2 2 What can be used 2 ORIGIN OF THE GAME even if the high school students are aware of these functions they would spoil the fun so we re not going to use them That leaves us with basic trigonometry inverse trigonometric functions log arithms square root taking number to a power factorial Are they powerful enough to solve any license plate We ll see Part II The Implementation The game en Score TEP Sam durak Landau Lifshitz Time 28 Round 1 D 5 7 Resign round Ask Tamm Help OK Quit match z du sin cos tan exp 5 asin acos atan In 1 Io lb sqrt premen CE Lefts
8. DAU user manual Harun Siljak November 9 2011 If you want to make a quick start with Dau don t read the entire manual simply read the help file i e part II of the manual The manual is here to tell you more about the game the characters history and mathematics Dau the game Figure 1 In case you were interested the license plates are from the Estonian SSR CONTENTS CONTENTS Contents I The History 1 Characters involved 11 Lev Davidovich Landau 1 2 Evgeny Mikhailovich Lifshitz 13 Igor Yevgenyevich Tamm 2 Origin of the game 2 Thegame is DORI a 2 ge 4 0 0 4 He XR Se Re imp dh 3I 39e Pride c8 OE Of OE ot e a 2 25 Whe can bessed socre oe x Sesam LARA eb II The Implementation 3 Interface 3 1 Playing a single player game aooaa 3 2 Playing a multiplayer game lll 4 Difficulty levels 4 1 Easy level 3 32 2 08 Sam Mrd Ru FIRE 42 Hard Level es toh ee Eda Bee ES 4 35 Terminiin ode de dera ds om Gy Boot e metes ee III The Mathematics 5 General solutions 5 1 Iterated trigonometry s sosoo oe e e 9 2 Factorials and trigonometry lll Do Binary Jogarib tyr 2e bekomm RR Ab en 6 Minimizing the functions set 6 1 Easy level of the game 2 2 0 2005 2a ee 6 2 Hard level of the game 2 0 0008 ee eee 6 9 The Teormmmum xong ow o 44 2 ae owed lern 10 11 12 15 19 19 19
9. Leftside editing 04 Figure 19 One player s screen in a multiplayer game en Score Your opponent N aam up this round 0 Landau Lifshitz Time 92 Round 4 Resign round Help OK Quit match i EU NEN sin cos tan 127 0 0 1 Just showing 2 4 5 asin acos atan You That chat works ME 2 3 aa b sart ste Send CE Leftside editing 04 Figure 20 Other player s screen in the same game 18 4 DIFFICULTY LEVELS 4 Difficulty levels Most games have certain difficulty levels DAU has three easy hard and teorminimum Teorminimum could be considered hardest although some may claim that hard level is somewhat more difficult than teorminimum 4 1 Easy level Basically all functions which we discussed in section 2 2 of this manual are allowed muParser is used for parsing of expressions all used functions except for factorial are implemented in muParser factorial and later double factorial are implemented in in DAU separately As you can see in figure 12 this means you can use addition substraction multiplication division take x to power y put decimal point between digits place brackets take square root exponentiate e take logarithm base 10 lg base e In and base 2 Ib sine cosine tangent inverse sine inverse cosine inverse tangent factorial 4 2 Hard Level In the hard level trigonometry is not allowed Expo
10. e top to bottom a status bar display of current license plate boxes for entering mathematical expressions button for confirmation of entry and a calculator pad with a double meaning one is possibility of entering digits functions and operations via that pad in boxes and the other one is showing what can be used in the round digits not appearing on the plate are automatically disabled and so are the functions operations which are not allowed at certain difficulty level Mathematical expressions can also be entered in boxes using the real keyboard most players will probably do so since it s faster If you enter a valid solution a green message of approval appears in the status bar and new round starts or the game ends if it was the last round If solution is invalid you will get a red message describing the type of mistake whether you used additional digits illegal functions changed digit order or simply made a typing error 3 2 Playing a multiplayer game As you might have expected multiplayer game has a lot of in common with single player game Options dialog is the same as before but additional options are available as you can see in figure 17 It s worth mentioning that in a multiplayer game computer of the player who chooses Landau as the playing character takes the role of game server while Lifshitz takes the role of game client that s just a technical note without any practical effect on the game itself
11. evels Your character since DAU is designed as a two player game you may choose what character will you be Landau default or Lifshitz It makes no difference in the single player game just tells the computer whose score should it increase after every correct answer After confirmation of preferred options game screen appears Depending on the rules chosen it may look different figures 12 13 14 Left corner box tracks score shows time elapsed in current round and what is the current round Buttons under it are the in game menu e Resign round skips the current round you don t get points for the round skipped New round starts if it wasn t the last round and if it was the game ends e Ask Tamm gives you a solution for current round using binary logarithm a window appears as shown in figure 15 This option is enabled only on easy level in single player since it is considered warm up mode of play e Help opens this part of user manual just like the Help button in the main menu e Quit match ends the game final score displays in a window figure 16 Same window appears when the game ends in its natural course i e when the last round ends 12 3 1 Playing a single player game 3 INTERFACE Set options Game options Number of rounds Unlimited 4 Time per round sec Unlimited Level of difficulty Easy A Your character Landau Figure 11 Single player options
12. from 5 1 That formula does not have to be iterated too many times in light of Dobganchuk s analysis References 1 B S Gorobets Krug Landau Letnii Sad Moskva 2006 2 M I Kaganov Landau s license plate game Quantum Vol 3 No 4 1983 3 H iljak Landau s game solution for the 21st century submitted to Matematicko fizicki list Zagreb 2011 23
13. ideediting 2 04 Figure 9 There is an Easter egg in the game and I couldn t help myself I had to mention it in this manual Anecdote says that Landau kept repeating silently during one of Paul Dirac s lectures Dirac fool durak until he finished with the lecture At the end Dirac turned to Dau and said Sam durak You re a fool yourself 10 3 INTERFACE Dau 1 0 Main Menu Dau the game The story of Dau user manual Single player game Multip layer game Help About Quit game Figure 10 DAu s main window 3 Interface In case you skipped the first part of the manual DAU is a computerized version of Landau s license plate game You get two pairs of digits separated with a hyphen e g 12 34 Goal is to manipulate digits by using standard elementary functions and mathematical operations to make an equality for instance 1 2 3 4or1x2 3 4or1 3 YA All given digits must be used exactly once no permutations allowed no additional digits allowed either You could easily play Landau s game walking on the street don t play and drive though but I wanted to make a PC game out of it so you can play it online with your friends or offline in your room without having to find real cars Choice is yours as shown in the figure 10 As you may see it is pretty self explainable The story of DAU button opens this manual while the Help button opens just a part of it namely this part
14. ion You argue that if we will cut off the ear of a cow and the ear of its offspring and so on sooner or later the earless cows will start to be born Yes that s right Lysenko replied Then Landau continued how do you explain the virgins that are still being born 1 2 Evgeny Mikhailovich Lifshitz 1 CHARACTERS INVOLVED L PEER N 43 Dai wx M d HJ Deltas 1 K Eor ir dw iy diaa z NAELU 33 l aa Mou MO beak i hfe Now 17 Eala 9 sais 37 i Dmi S 1 Uim dupuy iSi FE EMT TI M 7 4 I des andan s M daiala 3 Aruusep OS ee i Meo 5 7 loppa 35 ommo EE q Ik yp jaa fy n 1 a ro imas ds Mucca 3 5 r bas sI 8 a Gia pts Figure 4 Landau keeping track of Teorminimum students 1 2 Evgeny Mikhailovich Lifshitz Evgeny Mikhailovich Lifshitz Kharkiv 1915 Moscow 1985 Soviet physicist Academician Stalin and Lenin prize laureate His brother was Ilya Mikhailovich Lifshitz another well known physicist Coauthored the Course of Theoretical Physics with Landau one of the first people who passed Landau s Teorminimum made important contributions to general relativity phase transitions and other branches of theoretical physics On one of Bohr s visits to the USSR figure 6 he was asked how he had succeeded in making an excellent school of theoretical physics His answer was that he was never afraid to tell his students that he s a fool Lifshitz was the official translator and mistra
15. nd it according to 2 Gorobets in 1 noted that secant isn t in Russian high school curriculum anymore and therefore the solution isn t valid today In one of the articles from Nauka i zhizn reprinted in 1 S N Fedin gives an equivalent formula to the previous one but without secant which uses the fact that tan arccot x li yn 1 tan arccot cos arctan yn Note however that neither of these two variants can be used in DAU since cotangent inverse cotangent and secant are not available partly because they were not available in muParser partly because they don t put them on calcula tors these days That doesn t mean the formulae are completely inapplicable think of another way to make a reciprocal I m not going to give away those small tricks find them yourself I Dobganchuk has shown that using x Ub lg log and factorial one can either solve a license plate or reduce it to one of three forms 0 V3 0 V2 or V2 V3 as noted in 1 5 2 Factorials and trigonometry You may know by heart that 6 720 Since 720 2 360 n is divisible by 360 for all n gt 6 This means that sinn 0 for such n and that gives us a simple idea we can reduce any two digits down to 0 if one is zero we simply multiply them if none is zero we use this trigonometric identity and obtain trivial 0 0 equality This solution was also introduced by Gorobets As you could have guessed you can t apply thi
16. nentiation e logarithms other than lg as well as decimal point placing also cannot be used On the other hand use of double factorial is allowed figure 13 4 3 Teorminimum Add trigonometry to hard level but take away double factorial addition mul tiplication division rising to power and you get Teorminimum figure 14 19 Part III The Mathematics Figure 21 In case you wondered what was the logo that appears in the game on the calculator pad it is a tribute to the first Soviet pocket calculator Elektronika B3 04 20 5 GENERAL SOLUTIONS 5 General solutions As we pointed out before there are general solutions which you can use to solve any license plate combination Three of those will be presented here What is the point of such solutions On one hand they ruin the fun of the game since you don t have to use your imagination anymore while on the other hand looking for new general solutions gives a totally new dimension to the game 5 1 Iterated trigonometry Knowing that tan z 1 sin woos x sec x it is easy to make a function f n n 1 sec arctan yn Taking square roots in this equality yields vn 1 sec arctan yn Iterated use of this formula makes it possible to transform the smaller num ber on the plates into the larger one making the necessary equality hold This was the solution Kaganov presented to Landau after one of the Kharkiv math ematicians fou
17. nslated it saying He was never afraid to tell his students that they are fools Only after the academicians listening the lecture started laughing Lifshitz realized that he made a mistake and apologized Pyotr Kapitsa commented No Lifshitz you didn t make a mistake Exactly that is the difference between Bohr s and Landau s school of physics 1 3 Igor Yevgenyevich Tamm Igor Yevgenyevich Tamm Vladivostok 1895 Moscow 1971 Soviet physicist Academician Hero of Socialist Labour Stalin and Nobel prize laureate To gether with Pavel Cherenkov and Ilya Frank Tamm won the Nobel prize in 1958 for the discovery of Cherenkov radiation He worked on problems of ther monuclear reactors plasma preservation tokamak Tamm has nothing to do with Landau s game It s just that I loved that photo shown in figure 7 both Lifshitz and Landau are there Tamm explains something to them and there s even a car in the picture How convenient you 1 3 Igor Yevgenyevich Tamm 1 CHARACTERS INVOLVED Figure 5 Evgeny Mikhailovich Lifshitz Figure 6 Lev Landau and Niels Bohr 1 3 Igor Yevgenyevich Tamm 1 CHARACTERS INVOLVED Figure 7 Igor Yevgenyevich Tamm must admit My favourite anecdote about Tamm During the Russian revolution he was a physics professor at the University of Odessa in the Ukraine Food was in short supply so he made a trip to a nearby village in search of food While he was in the village a
18. one round e Your port Opponent s IP Opponent s port data needed for multi player connection Default port number for DAU is 31416 If there is a need to change it you can always do it in this dialog From the moment when player selects Multiplayer game in main menu port 31416 or the port s he selects starts waiting for invitations from other players Such invitations are sent by clicking the Offer rules button in this dialog As soon as a set of rules is offered to a player s he gets a message like in figure 18 where the other player is identified with IP address s he uses If the offer is accepted game begins on both computers If not the player who offered the rules is informed about the other player s refusal In game messages and interface are same as in single player game see fig ures 19 and 20 for an example It is worth mentioning that you are informed via status bar about every correct answer round skipping or match quitting done by your opponent If one of the players leaves the game other gets that information and game ends 17 3 2 Playing a multiplayer game 3 INTERFACE en E You skipped this round Waiting for your opponent Y 0 N Landau Lifshitz Time 82 Round 1 Resign round Tamm Help Quit match u gt sim cos tan You Just showing 4 5 i asin acos atan 127 0 0 1 That chat works ME 2 3 Wa Ni b sart te Send C CE
19. s general solution in DAU since implemented trigonometric functions work with radians I could have implemented alternative functions for angles in degrees but it would spoil the fun If anyone wants to use such functions s he can edit the source code 21 5 3 Binary logarithm 5 GENERAL SOLUTIONS 5 3 Binary logarithm The following solution is mine as introduced in 3 It uses the fact that logs 2 1 log 1 0 log 4 2 so it s easy to convert a 4 into a 2 a 2 into a 1 and 1 into a 0 That way one can make trivial 0 0 equalities whenever sum difference or quotient of a pair of digits is 0 1 2 or 4 Only 5 pairs of digits haven t got that property but they can be reduced to 2 1 0 using square root and or factorial How can we use log when we can t use additional digits and we need a 2 for logarithm s base there According to the ISO standard binary logarithm logarithm with base two is special just like logarithm with base 10 and with base e all three have special symbols lg for base 10 In for base e lb for base 2 This indicates the importance of these bases 10 and e earned their special place through centuries while base 2 became important in the second half of the last century with development of computers and theory of information This solution can be used in DAu only on the easy level though since ld is not allowed on upper levels As you may have noticed this is the solution you get when you Ask Tamm
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