The numbers $ 2, \ldots, 2015 $ are written on a blackboard. Each minute any two numbers $ x $ and $ y $ are wiped out and are replaced by two numbers $\displaystyle \frac { 4x + 3y } { 5 } $ and $\displaystyle \frac { 24x - 7y } { 25 } $.
Is it possible that after some time the numbers $ 1, \ldots, 2014 $ are written on the blackboard ?
Yes it is possible.
I started by creating a huge HTML table containing an overview of the generated values (right click the image and select "View image" for bigger version):
I ignored non integer values, therefore most of the fields are empty. The integer values form a regular pattern. The highlighted fields are the ones which keep one of the values unchanged. In addition to the 3 variants mentioned by GentlePurpleRain above, there is one more which uses negative values.
Using a highlighted red field it's possible to increase a single value while keeping the other unchanged. Using a highlighted blue field allows to decrease a single value. However, this is not enough to solve the puzzle. The next step was to increase the complexity. Look at the field representing [6, 17] -> [15, 1]. This changes two values, but we can restore one of them in a second step, namely [15 30] -> [30 6]. The effect of both these replacements is [17] -> [1].
There are more possibilities, e.g. [255, 510] -> [510, 102] and [102, 289] -> [255, 17] result in [289] -> [17]. Together with the previous combination we have [289] -> [1].
This can be continued further, which I have done using a computer program. It starts with the highlighted fields, and in each iteration adds more possible single number transformations to the list, until it finds the transformation [2015] -> [1]. See next chapter for the result.
Here is a possible sequence. It's not necessarily the shortest one, it's the one which was easiest to find.
Replacement Surplus Missing
. [2015] [1]
[300, 100] -> [300, 260] [260, 2015] [1, 100]
[180, 260] -> [300, 100] [300, 2015] [1, 180]
[300, 600] -> [600, 120] [120, 2015] [1, 180]
[135, 120] -> [180, 96] [96, 2015] [1, 135]
[96, 97] -> [135, 65] [65, 2015] [1, 97]
[120, 65] -> [135, 97] [135, 2015] [1, 120]
[96, 72] -> [120, 72] [135, 2015] [1, 96]
[135, 120] -> [180, 96] [180, 2015] [1, 120]
[252, 189] -> [315, 189] [180, 315, 2015] [1, 120, 252]
[315, 180] -> [360, 252] [360, 2015] [1, 120]
[360, 720] -> [720, 144] [144, 2015] [1, 120]
[167, 144] -> [220, 120] [220, 2015] [1, 167]
[308, 231] -> [385, 231] [220, 385, 2015] [1, 167, 308]
[385, 220] -> [440, 308] [440, 2015] [1, 167]
[440, 880] -> [880, 176] [176, 2015] [1, 167]
[176, 7] -> [145, 167] [145, 2015] [1, 7]
[375, 750] -> [750, 150] [145, 150, 2015] [1, 7, 375]
[150, 425] -> [375, 25] [25, 145, 2015] [1, 7, 425]
[450, 25] -> [375, 425] [145, 375, 2015] [1, 7, 450]
[375, 250] -> [450, 290] [145, 290, 2015] [1, 7, 250]
[290, 30] -> [250, 270] [145, 270, 2015] [1, 7, 30]
[225, 450] -> [450, 90] [90, 145, 270, 2015] [1, 7, 30, 225]
[90, 255] -> [225, 15] [15, 145, 270, 2015] [1, 7, 30, 255]
[270, 15] -> [225, 255] [145, 225, 2015] [1, 7, 30]
[225, 450] -> [450, 90] [90, 145, 2015] [1, 7, 30]
[270, 90] -> [270, 234] [145, 234, 2015] [1, 7, 30]
[390, 130] -> [390, 338] [145, 234, 338, 2015] [1, 7, 30, 130]
[234, 338] -> [390, 130] [145, 390, 2015] [1, 7, 30]
[145, 390] -> [350, 30] [350, 2015] [1, 7]
[315, 105] -> [315, 273] [273, 350, 2015] [1, 7, 105]
[189, 273] -> [315, 105] [315, 350, 2015] [1, 7, 189]
[224, 168] -> [280, 168] [280, 315, 350, 2015] [1, 7, 189, 224]
[315, 280] -> [420, 224] [350, 420, 2015] [1, 7, 189]
[420, 315] -> [525, 315] [350, 525, 2015] [1, 7, 189]
[350, 525] -> [595, 189] [595, 2015] [1, 7]
[595, 2015] -> [1685, 7] [1685] [1]
[60, 20] -> [60, 52] [52, 1685] [1, 20]
[36, 52] -> [60, 20] [60, 1685] [1, 36]
[55, 60] -> [80, 36] [80, 1685] [1, 55]
[112, 84] -> [140, 84] [80, 140, 1685] [1, 55, 112]
[140, 80] -> [160, 112] [160, 1685] [1, 55]
[160, 320] -> [320, 64] [64, 1685] [1, 55]
[64, 23] -> [65, 55] [65, 1685] [1, 23]
[65, 130] -> [130, 26] [26, 1685] [1, 23]
[26, 7] -> [25, 23] [25, 1685] [1, 7]
[120, 40] -> [120, 104] [25, 104, 1685] [1, 7, 40]
[72, 104] -> [120, 40] [25, 120, 1685] [1, 7, 72]
[120, 240] -> [240, 48] [25, 48, 1685] [1, 7, 72]
[89, 48] -> [100, 72] [25, 100, 1685] [1, 7, 89]
[100, 25] -> [95, 89] [95, 1685] [1, 7]
[35, 95] -> [85, 7] [85, 1685] [1, 35]
[85, 170] -> [170, 34] [34, 1685] [1, 35]
[34, 13] -> [35, 29] [29, 1685] [1, 13]
[168, 126] -> [210, 126] [29, 210, 1685] [1, 13, 168]
[210, 120] -> [240, 168] [29, 240, 1685] [1, 13, 120]
[336, 252] -> [420, 252] [29, 240, 420, 1685] [1, 13, 120, 336]
[420, 240] -> [480, 336] [29, 480, 1685] [1, 13, 120]
[480, 960] -> [960, 192] [29, 192, 1685] [1, 13, 120]
[181, 192] -> [260, 120] [29, 260, 1685] [1, 13, 181]
[500, 1000] -> [1000, 200] [29, 200, 260, 1685] [1, 13, 181, 500]
[475, 200] -> [500, 400] [29, 260, 400, 1685] [1, 13, 181, 475]
[400, 300] -> [500, 300] [29, 260, 500, 1685] [1, 13, 181, 475]
[500, 125] -> [475, 445] [29, 260, 445, 1685] [1, 13, 125, 181]
[260, 445] -> [475, 125] [29, 475, 1685] [1, 13, 181]
[525, 175] -> [525, 455] [29, 455, 475, 1685] [1, 13, 175, 181]
[455, 935] -> [925, 175] [29, 475, 925, 1685] [1, 13, 181, 935]
[475, 925] -> [935, 197] [29, 197, 1685] [1, 13, 181]
[197, 29] -> [175, 181] [175, 1685] [1, 13]
[256, 192] -> [320, 192] [175, 320, 1685] [1, 13, 256]
[360, 320] -> [480, 256] [175, 480, 1685] [1, 13, 360]
[480, 960] -> [960, 192] [175, 192, 1685] [1, 13, 360]
[431, 192] -> [460, 360] [175, 460, 1685] [1, 13, 431]
[196, 147] -> [245, 147] [175, 245, 460, 1685] [1, 13, 196, 431]
[245, 140] -> [280, 196] [175, 280, 460, 1685] [1, 13, 140, 431]
[280, 460] -> [500, 140] [175, 500, 1685] [1, 13, 431]
[500, 175] -> [505, 431] [505, 1685] [1, 13]
[505, 1685] -> [1415, 13] [1415] [1]
[600, 200] -> [600, 520] [520, 1415] [1, 200]
[360, 520] -> [600, 200] [600, 1415] [1, 360]
[600, 1200] -> [1200, 240] [240, 1415] [1, 360]
[445, 240] -> [500, 360] [500, 1415] [1, 445]
[500, 75] -> [445, 459] [459, 1415] [1, 75]
[405, 810] -> [810, 162] [162, 459, 1415] [1, 75, 405]
[162, 459] -> [405, 27] [27, 1415] [1, 75]
[86, 27] -> [85, 75] [85, 1415] [1, 86]
[75, 150] -> [150, 30] [30, 85, 1415] [1, 75, 86]
[30, 85] -> [75, 5] [5, 1415] [1, 86]
[140, 280] -> [280, 56] [5, 56, 1415] [1, 86, 140]
[133, 56] -> [140, 112] [5, 112, 1415] [1, 86, 133]
[112, 84] -> [140, 84] [5, 140, 1415] [1, 86, 133]
[140, 5] -> [115, 133] [115, 1415] [1, 86]
[115, 230] -> [230, 46] [46, 1415] [1, 86]
[103, 46] -> [110, 86] [110, 1415] [1, 103]
[280, 210] -> [350, 210] [110, 350, 1415] [1, 103, 280]
[350, 200] -> [400, 280] [110, 400, 1415] [1, 103, 200]
[400, 800] -> [800, 160] [110, 160, 1415] [1, 103, 200]
[130, 160] -> [200, 80] [80, 110, 1415] [1, 103, 130]
[80, 110] -> [130, 46] [46, 1415] [1, 103]
[120, 40] -> [120, 104] [46, 104, 1415] [1, 40, 103]
[72, 104] -> [120, 40] [46, 120, 1415] [1, 72, 103]
[110, 120] -> [160, 72] [46, 160, 1415] [1, 103, 110]
[224, 168] -> [280, 168] [46, 160, 280, 1415] [1, 103, 110, 224]
[280, 160] -> [320, 224] [46, 320, 1415] [1, 103, 110]
[320, 640] -> [640, 128] [46, 128, 1415] [1, 103, 110]
[128, 46] -> [130, 110] [130, 1415] [1, 103]
[112, 84] -> [140, 84] [130, 140, 1415] [1, 103, 112]
[140, 80] -> [160, 112] [130, 160, 1415] [1, 80, 103]
[130, 160] -> [200, 80] [200, 1415] [1, 103]
[600, 200] -> [600, 520] [520, 1415] [1, 103]
[520, 1415] -> [1265, 103] [1265] [1]
[120, 40] -> [120, 104] [104, 1265] [1, 40]
[72, 104] -> [120, 40] [120, 1265] [1, 72]
[168, 126] -> [210, 126] [120, 210, 1265] [1, 72, 168]
[210, 120] -> [240, 168] [240, 1265] [1, 72]
[145, 240] -> [260, 72] [260, 1265] [1, 145]
[364, 273] -> [455, 273] [260, 455, 1265] [1, 145, 364]
[455, 260] -> [520, 364] [520, 1265] [1, 145]
[520, 1265] -> [1175, 145] [1175] [1]
[750, 1500] -> [1500, 300] [300, 1175] [1, 750]
[300, 850] -> [750, 50] [50, 1175] [1, 850]
[900, 50] -> [750, 850] [750, 1175] [1, 900]
[750, 1500] -> [1500, 300] [300, 1175] [1, 900]
[1025, 300] -> [1000, 900] [1000, 1175] [1, 1025]
[1000, 2000] -> [2000, 400] [400, 1175] [1, 1025]
[400, 1175] -> [1025, 55] [55] [1]
[300, 600] -> [600, 120] [55, 120] [1, 300]
[285, 120] -> [300, 240] [55, 240] [1, 285]
[240, 155] -> [285, 187] [55, 187] [1, 155]
[105, 35] -> [105, 91] [55, 91, 187] [1, 35, 155]
[91, 187] -> [185, 35] [55, 185] [1, 155]
[55, 185] -> [155, 1] [] []
I believe that no.
For many pairs of numbers x and y, (4x + 3y) / 5 and (24x − 7y) / 25 won't even be integers. For example:
x = 2, y = 7:
(4x + 3y) / 5 = (8 + 21) / 5 = 29/5 = 5.8
(24x − 7y) / 25 = (48 - 49) / 25 = -1/25 = -0.04
In about only 1/5 of the steps the first expression will be an integer, and in about only 1/25 of the steps the second expression will be an integer, per divisibility rules, and assuming that the numerator is a random integer.
And, by construction, most non-integer numbers will continue to be operated into other non-integer numbers - with bigger denominators in fractional form.
The question can be turned around into: Find a way to generate all numbers from 1 to 2014, given the operations with x and y above, and choosing x and y for each step. Expecting it to happen at random is extremely improbable.
