[Pupułka]

Czołem,

kiedyś przeczytałem post JB chyba, na może FLD, w którym pisał -- tak mi się majaczyło -- że nagrody za kombosy są w jakiś sposób związane z wartością pierwiastka z czy potęgi dwójki. Wróciłem dzisiaj do tego zagadnienia, i nastąpił... nieskoordynowany szereg programistyczno-matematyczno-Google'owo-Wolfram|Alpha'owych rozważań, w wyniku których udało mi się ustalić... jeśli cokolwiek, to to, że wzór na liczbę pięter n-tej nagrody w Icy Tower to, w PHP-owym zapisie,

Kod:

round(pow(2, ($n - 8) / 2) * 100)

. Jestem laikiem, więc nie potrafię opisać, co dokładnie mu dolega. Pewnie jest niedokładny, niezredukowany do optymalnego zapisu, i tak dalej. Ale, caveat lector, może ktoś coś z tych rozważań zabawnego wyciągnie. Może wiedziałbym nieco lepiej, co tu zrobiłem, gdybym był w ostatnich dniach w lepszej formie.

W każdym razie poniższe pokazuje chyba kilka ciekawych rzeczy, w sensie liczbowym.

Kod:

20:32        *** Zl joined #freelunchdesign         Free Lunch Design's official channel "old" edition | Visit us at http://www.freelunchdesign.com/forum/ | Old topic is old.         Topic set by JohnBeak!~John.Beak@13.121.broadband5.iol.cz on Mon May 26 2014 12:41:40 GMT+0200 (CEST) 20:33    Zl    What's the formula for the amount of floors required by the nth reward again? 20:33    Zl    int floors_required(int nth_reward_type) 20:37    Zl    Ah wait, I think I remembered (from when it was mentioned on FLD last time). 20:38    Zl    I think it's sqrt(2) * floors_required(nth_reward_type - 1) . 20:39    Zl    ...Now I'm trying to think how to define it non-recursively. >_< 20:39    Zl    Wah why am I such an abysmal programmer. 20:42    Zl    *puts on small voice* Help. 20:44    Zl    Wait... 20:46    Zl    1 = 4 20:46    Zl    2 = 4*1.41 20:46    Zl    r = 1.41 20:47    Zl    3 = (3r)r = 3r*r2 20:48    Zl    4 = (3r*r^2)r = (???)*r^3 20:48    Zl    n = (???)*r^n 20:49    Zl    I made mistakes in this even. >_< 20:55    Zl    http://www.wolframalpha.com/input/?i=x(n)+=+sqrt(2)+*+x(n+-+1) 20:55    Zl    I'm not sure I understand the output. 21:09    Zl    Okay, wait. 21:10    Zl    After mad guessing I arrived at 2^((n+5)/2)*1.1. 21:10    Zl    * 2^((n+5)/2)*1.1 21:10    Zl    Inaccurate for lower n. 21:12    Zl    <?php for($i = 1; $i <= 20; $i++) printf("%dth combo reward = from %d floors\n", $i, round(1.1 * pow(2, ($i + 5) / 2))); 21:14    Zl    Actually, the coefficient should be 1.105, not 1.1. 21:15    Zl    Or, fuck, even better, 1.10485. 21:15    Zl    This way the 14th reward is 800 and the 20th, 6400. 21:16    Zl    Well, 1.10485 it is then. Nice magic number. 21:17    Zl    <?php for($i = 1, $m = 1.10485, $n = 20; $i <= $n; $i++) printf("%dth combo reward = from %d floors\n", $i, round($m * pow(2, ($i + 5) / 2))); 21:17    Zl    http://pastebin.com/WcXLn5Sm 21:18    Zl    FUUUUUCK. 21:19    Zl    Now I caught myself trying to think of the algorithm to determine $m to the best possible value. >_< 21:19    Zl    It never ends. ;_; 21:21    Zl    Wait... 21:26    Zl    Wah. It's 1.1048543456 exactly, at least as far as PHP's precision allows to tell. 21:26    Zl    But I still don't understand the significance of that number. ;_; 21:26    Zl    But it shows in Google!... https://www.google.com/search?q="1.1048543456" 21:28    Zl    <?php for($i = 1, $m = 1.104854345605, $n = 30; $i <= $n; $i++) printf("%dth combo reward = from %f floors\n", $i, $m * pow(2, ($i + 5) / 2)); 21:28    Zl    Well, at least I know the 30th Icy Tower reward should have exactly 204800 floors. 21:28    Zl    Heh. 21:29    Zl    Knowing that Icy Tower only supports a limited number of floors, owing to the max value of the datatype, this could serve to infer the max number of rewards Icy Tower can possibly support. 21:29    Zl    Because certain rewards, rewards which ought to belong to a certain number of floors, will never be reachable. 21:30    Zl    ...Now what's that number. 21:30    Zl    "static final double TWO_TO_NEGATIVE_HALF_POW" 21:31    Zl    Hmmmmmm. 21:33    Zl    static final double TWO_TO_NEGATIVE_HALF_POW[] = { 1.0000000000E+00f, 7.0710678119E-01f, 5.0000000000E-01f, 3.5355339059E-01f, 2.5000000000E-01f, 1.7677669530E-01f, 1.2500000000E-01f, 8.8388347648E-02f, 6.2500000000E-02f, 4.4194173824E-02f, 3.1250000000E-02f, 2.2097086912E-02f, 1.5625000000E-02f, 1.1048543456E-02f, 21:33    Zl    I don't understaaaaaaaaand. 21:36    Zl    2^(-.5) is 7.07106... and so on. 21:36    Zl    But what's the formula for my 1.10485...? 21:39    Zl    Got it!... Kind of. 21:39    Zl    2^(13*-.5) 21:41    Zl    Okay. 21:42    Zl    The magic combo number is pow(2, 13 * -.5) * 100 . 21:42    Zl    But now, where did the 13 come from? ;_; 21:43    Zl    <?php for($i = 1, $m = pow(2, 13 * -.5) * 100, $n = 100; $i <= $n; $i++) printf("%03dth combo reward = from %f floors\n", $i, $m * pow(2, ($i + 5) / 2)); 21:43    Zl    http://pastebin.com/JKw3Nj5U 21:44    Zl    096th combo reward = from 1759218604441600.250000 floors 21:44    Zl    097th combo reward = from 2487910809580380.000000 floors 21:44    Zl    098th combo reward = from 3518437208883200.500000 floors 21:44    Zl    099th combo reward = from 4975821619160760.000000 floors 21:44    Zl    100th combo reward = from 7036874417766401.000000 floors 21:44    Zl    Math is bizarre. 21:46    Zl    Oh well. 21:48    Zl    In the final iteration of the code, there are still two numbers that to me look arbitrary, and which perhaps should somehow be reduced to just one single value: the 13 in the definition of the magic value, and 5 in the floor threshold calculation. But it's good enough, even if I don't understand at all what I've been doing for the last hour. 21:54    Zl    All right, final version, to i.a. circumvent some apparent printf limitations: 21:54    Zl    <?php for($i = 1, $m = pow(2, 13 * -.5), $n = 100; $i <= $n; $i++) printf("%03dth combo reward = from %.0f floors\n", $i, round(100 * $m * pow(2, ($i + 5) / 2))); 21:54    Zl    The mystery of the 13 I am leaving for anyone else to solve. 21:55    Zl    I am happy enough to have persisted inquisitively enough to restate the "magic value" into an expression. 21:56    Zl    http://pastebin.com/4CHp11ZN 21:59    Zl    Wah, 13 * -.5 could be written -6.5 obviously. 22:01    Zl    All right, I'll post this on Icy.pl. Maybe it will entertain somebody.

Kod:

<?php $reward = function($n) {     return round(pow(2, ($n - 8) / 2) * 100); }; for($i = 1; $i <= 2 << 6; $i++) {     printf("% 3dth combo reward % 30.0f floors\n", $i, $reward($i)); } ?>

Kod:

>php -f "combo.php"   1th combo reward                              9 floors   2th combo reward                             13 floors   3th combo reward                             18 floors   4th combo reward                             25 floors   5th combo reward                             35 floors   6th combo reward                             50 floors   7th combo reward                             71 floors   8th combo reward                            100 floors   9th combo reward                            141 floors  10th combo reward                            200 floors  11th combo reward                            283 floors  12th combo reward                            400 floors  13th combo reward                            566 floors  14th combo reward                            800 floors  15th combo reward                           1131 floors  16th combo reward                           1600 floors  17th combo reward                           2263 floors  18th combo reward                           3200 floors  19th combo reward                           4525 floors  20th combo reward                           6400 floors  21th combo reward                           9051 floors  22th combo reward                          12800 floors  23th combo reward                          18102 floors  24th combo reward                          25600 floors  25th combo reward                          36204 floors  26th combo reward                          51200 floors  27th combo reward                          72408 floors  28th combo reward                         102400 floors  29th combo reward                         144815 floors  30th combo reward                         204800 floors  31th combo reward                         289631 floors  32th combo reward                         409600 floors  33th combo reward                         579262 floors  34th combo reward                         819200 floors  35th combo reward                        1158524 floors  36th combo reward                        1638400 floors  37th combo reward                        2317048 floors  38th combo reward                        3276800 floors  39th combo reward                        4634095 floors  40th combo reward                        6553600 floors  41th combo reward                        9268190 floors  42th combo reward                       13107200 floors  43th combo reward                       18536380 floors  44th combo reward                       26214400 floors  45th combo reward                       37072760 floors  46th combo reward                       52428800 floors  47th combo reward                       74145520 floors  48th combo reward                      104857600 floors  49th combo reward                      148291040 floors  50th combo reward                      209715200 floors  51th combo reward                      296582080 floors  52th combo reward                      419430400 floors  53th combo reward                      593164160 floors  54th combo reward                      838860800 floors  55th combo reward                     1186328320 floors  56th combo reward                     1677721600 floors  57th combo reward                     2372656641 floors  58th combo reward                     3355443200 floors  59th combo reward                     4745313281 floors  60th combo reward                     6710886400 floors  61th combo reward                     9490626562 floors  62th combo reward                    13421772800 floors  63th combo reward                    18981253125 floors  64th combo reward                    26843545600 floors  65th combo reward                    37962506250 floors  66th combo reward                    53687091200 floors  67th combo reward                    75925012499 floors  68th combo reward                   107374182400 floors  69th combo reward                   151850024999 floors  70th combo reward                   214748364800 floors  71th combo reward                   303700049998 floors  72th combo reward                   429496729600 floors  73th combo reward                   607400099995 floors  74th combo reward                   858993459200 floors  75th combo reward                  1214800199990 floors  76th combo reward                  1717986918400 floors  77th combo reward                  2429600399981 floors  78th combo reward                  3435973836800 floors  79th combo reward                  4859200799962 floors  80th combo reward                  6871947673600 floors  81th combo reward                  9718401599923 floors  82th combo reward                 13743895347200 floors  83th combo reward                 19436803199847 floors  84th combo reward                 27487790694400 floors  85th combo reward                 38873606399693 floors  86th combo reward                 54975581388800 floors  87th combo reward                 77747212799387 floors  88th combo reward                109951162777600 floors  89th combo reward                155494425598774 floors  90th combo reward                219902325555200 floors  91th combo reward                310988851197548 floors  92th combo reward                439804651110400 floors  93th combo reward                621977702395095 floors  94th combo reward                879609302220800 floors  95th combo reward               1243955404790190 floors  96th combo reward               1759218604441600 floors  97th combo reward               2487910809580380 floors  98th combo reward               3518437208883200 floors  99th combo reward               4975821619160760 floors 100th combo reward               7036874417766400 floors 101th combo reward               9951643238321520 floors 102th combo reward              14073748835532800 floors 103th combo reward              19903286476643040 floors 104th combo reward              28147497671065600 floors 105th combo reward              39806572953286080 floors 106th combo reward              56294995342131200 floors 107th combo reward              79613145906572160 floors 108th combo reward             112589990684262400 floors 109th combo reward             159226291813144320 floors 110th combo reward             225179981368524800 floors 111th combo reward             318452583626288640 floors 112th combo reward             450359962737049600 floors 113th combo reward             636905167252577280 floors 114th combo reward             900719925474099200 floors 115th combo reward            1273810334505154560 floors 116th combo reward            1801439850948198400 floors 117th combo reward            2547620669010309120 floors 118th combo reward            3602879701896396800 floors 119th combo reward            5095241338020618240 floors 120th combo reward            7205759403792793600 floors 121th combo reward           10190482676041236480 floors 122th combo reward           14411518807585587200 floors 123th combo reward           20380965352082472960 floors 124th combo reward           28823037615171174400 floors 125th combo reward           40761930704164945920 floors 126th combo reward           57646075230342348800 floors 127th combo reward           81523861408329891840 floors 128th combo reward          115292150460684697600 floors >Exit code: 0

[Pupułka]

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