9
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During our latest space mission to Mars, humankind has retrieved a curious alien circuit. Our historians think it must be a calendar of some sort.

enter image description here

After speaking with some alien linguists, we know now that the characters are symbols of counting and we deduced that they must be numbers.

enter image description here

When the engineering deparment hooked up the circuit to a low-frequency signal, it started moving, emitting light and spitting out similar symbols. Freely translated, the engineers observed and wrote down the following sequence

[2, 1], [5, 5], [8, 2], [11, 1], [14, 5], [15, 4], [18, 3], [21, 2], [22, 3], [25, 2], [28, 1], [31, 5], [34, 2], [37, 1], [40, 5], [43, 2], [46, 1], [49, 5], [50, 4], [51, 2], [52, 3], [55, 2], [58, 1], [61, 5], [64, 2], [67, 1], [70, 5], [71, 4], [74, 3], [77, 2], [80, 1], [81, 2], [82, 3], [85, 2], [88, 1], [91, 5], [92, 4], [93, 2], [94, 3], [97, 2], [100, 1], [101, 2], [104, 1], [107, 5], [110, 2], [113, 1], [116, 5], [119, 2], [122, 1], [125, 5], [128, 2], [131, 1], [134, 5], [137, 2], [140, 1], [141, 2], [142, 3], [145, 2], [148, 1], [151, 5], [154, 2], [157, 1], [160, 5], [163, 2], [166, 1], [169, 5], [172, 2], [175, 1], [176, 2], [179, 1], [182, 5], [183, 4], [186, 3], [189, 2], [190, 3], [191, 5], [194, 2], [197, 1], [200, 5], [203, 2], [206, 1], [209, 5], [212, 2], [215, 1], [216, 2], [217, 3], [220, 2], [223, 1], [226, 5], [229, 2], [232, 1], [235, 5], [238, 2], [241, 1], [244, 5], [247, 2], [250, 1], [251, 2], [254, 1], [257, 5], [260, 2], [263, 1], [266, 5], [267, 4], [270, 3], [271, 5], [274, 2], [277, 1], [280, 5], [281, 4], [284, 3], [287, 2], [290, 1], [291, 2], [292, 3], [295, 2], [298, 1], [301, 5], [302, 4], [303, 2], [304, 3], [307, 2], [310, 1], [311, 2], [314, 1], [317, 5], [320, 2], [323, 1], [326, 5], [329, 2], [332, 1], [335, 5], [338, 2], [341, 1], [344, 5], [347, 2], [350, 1], [351, 2], [352, 3], [355, 2], [358, 1], [361, 5], [364, 2], [367, 1], [370, 5], [373, 2], [376, 1], [379, 5], [382, 2], [385, 1], [386, 2], [389, 1], [392, 5], [393, 4], [396, 3], [399, 2], [400, 3], [401, 5], [404, 2], [407, 1], [410, 5], [413, 2], [416, 1], [419, 5], [422, 2], [425, 1], [426, 2], [427, 3], [430, 2], [433, 1], [436, 5], [439, 2], [442, 1], [445, 5], [448, 2], [451, 1], [454, 5], [457, 2], [460, 1], [461, 2], [464, 1], [467, 5], [470, 2], [473, 1], [476, 5], [477, 4], [480, 3], [481, 5], [484, 2], [487, 1], [490, 5], [491, 4], [494, 3], [497, 2], [500, 1], [501, 2], [502, 3], [505, 2], [508, 1], [511, 5], [512, 4], [513, 2], [514, 3], [517, 2], [520, 1], [521, 2], [524, 1], [527, 5], [530, 2], [533, 1], [536, 5], [539, 2], [542, 1], [545, 5], [548, 2], [551, 1], [554, 5], [557, 2], [560, 1], [561, 2], [562, 3], [565, 2], [568, 1], [571, 5], [574, 2], [577, 1], [580, 5], [583, 2], [586, 1], [589, 5], [592, 2], [595, 1], [596, 2], [599, 1], [602, 5], [603, 4], [606, 3], [609, 2], [610, 3], [611, 5], [614, 2], [617, 1], [620, 5], [623, 2], [626, 1], [629, 5], [632, 2], [635, 1], [636, 2], [637, 3], [640, 2], [643, 1], [646, 5], [649, 2], [652, 1], [655, 5], [658, 2], [661, 1], [664, 5], [667, 2], [670, 1], [671, 2], [674, 1], [677, 5], [680, 2], [683, 1], [686, 5], [687, 4], [690, 3], [691, 5], [694, 2], [697, 1], [700, 5], [701, 4], [704, 3], [707, 2], [710, 1], [711, 2], [712, 3], [715, 2], [718, 1], [721, 5], [722, 4], [723, 2], [724, 3], [727, 2], [730, 1], [731, 2], [734, 1], [737, 5], [740, 2], [743, 1], [746, 5], [749, 2], [752, 1], [755, 5], [758, 2], [761, 1], [764, 5], [767, 2], [770, 1], [771, 2], [772, 3], [775, 2], [778, 1], [781, 5], [784, 2], [787, 1], [790, 5], [793, 2], [796, 1], [799, 5], [802, 2], [805, 1], [806, 2], [809, 1], [812, 5], [813, 4], [816, 3], [819, 2], [820, 3], [821, 5], [824, 2], [827, 1], [830, 5], [833, 2], [836, 1], [839, 5], [842, 2], [845, 1], [846, 2], [847, 3], [850, 2], [853, 1], [856, 5], [859, 2], [862, 1], [865, 5], [868, 2], [871, 1], [874, 5], [877, 2], [880, 1], [881, 2], [884, 1], [887, 5], [890, 2], [893, 1], [896, 5], [897, 4], [900, 3], [901, 5], [904, 2], [907, 1], [910, 5], [911, 4], [914, 3], [917, 2], [920, 1], [921, 2], [922, 3], [925, 2], [928, 1], [931, 5], [932, 4], [933, 2], [934, 3], [937, 2], [940, 1], [941, 2], [944, 1], [947, 5], [950, 2], [953, 1], [956, 5], [959, 2], [962, 1], [965, 5], [968, 2], [971, 1], [974, 5], [977, 2], [980, 1], [981, 2], [982, 3], [985, 2], [988, 1], [991, 5], [994, 2], [997, 1], [1000, 5], [1003, 2], [1006, 1], [1009, 5], [1012, 2], [1015, 1], [1016, 2], [1019, 1], [1022, 5], [1023, 4], [1026, 3], [1029, 2], [1030, 3], [1031, 5], [1034, 2], [1037, 1], [1040, 5], [1043, 2], [1046, 1], [1049, 5], [1052, 2], [1055, 1], [1056, 2], [1057, 3], [1060, 2], [1063, 1], [1066, 5], [1069, 2], [1072, 1], [1075, 5], [1078, 2], [1081, 1], [1084, 5], [1087, 2], [1090, 1], [1091, 2], [1094, 1], [1097, 5], [1100, 2], [1103, 1], [1106, 5], [1107, 4], [1110, 3], [1111, 5], [1114, 2], [1117, 1], [1120, 5], [1121, 4], [1124, 3], [1127, 2], [1130, 1], [1131, 2], [1132, 3], [1135, 2], [1138, 1], [1141, 5], [1142, 4], [1143, 2], [1144, 3], [1147, 2], [1150, 1], [1151, 2], [1154, 1], [1157, 5], [1160, 2], [1163, 1], [1166, 5], [1169, 2], [1172, 1], [1175, 5], [1178, 2], [1181, 1], [1184, 5], [1187, 2], [1190, 1], [1191, 2], [1192, 3], [1195, 2], [1198, 1], [1201, 5], [1204, 2], [1207, 1], [1210, 5], [1213, 2], [1216, 1], [1219, 5], [1222, 2], [1225, 1], [1226, 2], [1229, 1], [1232, 5], [1233, 4], [1236, 3], [1239, 2], [1240, 3], [1241, 5], [1244, 2], [1247, 1], [1250, 5], [1253, 2], [1256, 1], [1259, 5], [1262, 2], [1265, 1], [1266, 2], [1267, 3], [1270, 2], [1273, 1], [1276, 5], [1279, 2], [1282, 1], [1285, 5], [1288, 2], [1291, 1], [1294, 5], [1297, 2], [1300, 1], [1301, 2], [1304, 1], [1307, 5], [1310, 2], [1313, 1], [1316, 5], [1317, 4], [1320, 3], [1321, 5], [1324, 2], [1327, 1], [1330, 5], [1331, 4], [1334, 3], [1337, 2], [1340, 1], [1341, 2], [1342, 3], [1345, 2], [1348, 1], [1351, 5], [1352, 4], [1353, 2], [1354, 3], [1357, 2], [1360, 1], [1361, 2], [1364, 1], [1367, 5], [1370, 2], [1373, 1], [1376, 5], [1379, 2], [1382, 1], [1385, 5], [1388, 2], [1391, 1], [1394, 5], [1397, 2], [1400, 1], [1401, 2], [1402, 3], [1405, 2], [1408, 1], [1411, 5], [1414, 2], [1417, 1], [1420, 5], [1423, 2], [1426, 1], [1429, 5], [1432, 2], [1435, 1], [1436, 2], [1439, 1], [1442, 5], [1443, 4], [1446, 3], [1449, 2], [1450, 3], [1451, 5], [1454, 2], [1457, 1], [1460, 5], [1463, 2], [1466, 1], [1469, 5], [1472, 2], [1475, 1], [1476, 2], [1477, 3], [1480, 2], [1483, 1], [1486, 5], [1489, 2], [1492, 1], [1495, 5], [1498, 2], [1501, 1], [1504, 5], [1507, 2], [1510, 1], [1511, 2], [1514, 1], [1517, 5], [1520, 2], [1523, 1], [1526, 5], [1527, 4], [1530, 3], [1531, 5], [1534, 2], [1537, 1], [1540, 5], [1541, 4], [1544, 3], [1547, 2], [1550, 1], [1551, 2], [1552, 3], [1555, 2], [1558, 1], [1561, 5], [1562, 4], [1563, 2], [1564, 3], [1567, 2], [1570, 1], [1571, 2], [1574, 1], [1577, 5], [1580, 2], [1583, 1], [1586, 5], [1589, 2], [1592, 1], [1595, 5], [1598, 2], [1601, 1], [1604, 5], [1607, 2], [1610, 1], [1611, 2], [1612, 3], [1615, 2], [1618, 1], [1621, 5], [1624, 2], [1627, 1], [1630, 5], [1633, 2], [1636, 1], [1639, 5], [1642, 2], [1645, 1], [1646, 2], [1649, 1], [1652, 5], [1653, 4], [1656, 3], [1659, 2], [1660, 3], [1661, 5], [1664, 2], [1667, 1], [1670, 5], [1673, 2], [1676, 1], [1679, 5], [1682, 2], [1685, 1], [1686, 2], [1687, 3], [1690, 2], [1693, 1], [1696, 5], [1699, 2], [1702, 1], [1705, 5], [1708, 2], [1711, 1], [1714, 5], [1717, 2], [1720, 1], [1721, 2], [1724, 1], [1727, 5], [1730, 2], [1733, 1], [1736, 5], [1737, 4], [1740, 3], [1741, 5], [1744, 2], [1747, 1], [1750, 5], [1751, 4], [1754, 3], [1757, 2], [1760, 1], [1761, 2], [1762, 3], [1765, 2], [1768, 1], [1771, 5], [1772, 4], [1773, 2], [1774, 3], [1777, 2], [1780, 1], [1781, 2], [1784, 1], [1787, 5], [1790, 2], [1793, 1], [1796, 5], [1799, 2], [1802, 1], [1805, 5], [1808, 2], [1811, 1], [1814, 5], [1817, 2], [1820, 1], [1821, 2], [1822, 3], [1825, 2], [1828, 1], [1831, 5], [1834, 2], [1837, 1], [1840, 5], [1843, 2], [1846, 1], [1849, 5], [1852, 2], [1855, 1], [1856, 2], [1859, 1], [1862, 5], [1863, 4], [1866, 3], [1869, 2], [1870, 3], [1871, 5], [1874, 2], [1877, 1], [1880, 5], [1883, 2], [1886, 1], [1889, 5], [1892, 2], [1895, 1], [1896, 2], [1897, 3], [1900, 2], [1903, 1], [1906, 5], [1909, 2], [1912, 1], [1915, 5], [1918, 2], [1921, 1], [1924, 5], [1927, 2], [1930, 1], [1931, 2], [1934, 1], [1937, 5], [1940, 2], [1943, 1], [1946, 5], [1947, 4], [1950, 3], [1951, 5], [1954, 2], [1957, 1], [1960, 5], [1961, 4], [1964, 3], [1967, 2], [1970, 1], [1971, 2], [1972, 3], [1975, 2], [1978, 1], [1981, 5], [1982, 4], [1983, 2], [1984, 3], [1987, 2], [1990, 1], [1991, 2], [1994, 1], [1997, 5], [2000, 2], [2003, 1], [2006, 5], [2009, 2], [2012, 1], [2015, 5], [2018, 2], [2021, 1], [2024, 5], [2027, 2], [2030, 1], [2031, 2], [2032, 3], [2035, 2], [2038, 1], [2041, 5], [2044, 2], [2047, 1], [2050, 5], [2053, 2], [2056, 1], [2059, 5], [2062, 2], [2065, 1], [2066, 2], [2069, 1], [2072, 5], [2073, 4], [2076, 3], [2079, 2], [2080, 3], [2081, 5], [2084, 2], [2087, 1], [2090, 5], [2093, 2], [2096, 1], [2099, 5], [2102, 2], [2105, 1], [2106, 2], [2107, 3], [2110, 2], [2113, 1], [2116, 5], [2119, 2], [2122, 1], [2125, 5], [2128, 2], [2131, 1], [2134, 5], [2137, 2], [2140, 1], [2141, 2], [2144, 1], [2147, 5], [2150, 2], [2153, 1], [2156, 5], [2157, 4], [2160, 3], [2161, 5], [2164, 2], [2167, 1], [2170, 5], [2171, 4], [2174, 3], [2177, 2], [2180, 1], [2181, 2], [2182, 3], [2185, 2], [2188, 1], [2191, 5], [2192, 4], [2193, 2], [2194, 3], [2197, 2], [2200, 1], [2201, 2], [2204, 1], [2207, 5], [2210, 2], [2213, 1], [2216, 5], [2219, 2], [2222, 1], [2225, 5], [2228, 2], [2231, 1], [2234, 5], [2237, 2], [2240, 1], [2241, 2], [2242, 3], [2245, 2], [2248, 1], [2251, 5], [2254, 2], [2257, 1], [2260, 5], [2263, 2], [2266, 1], [2269, 5], [2272, 2], [2275, 1], [2276, 2], [2279, 1], [2282, 5], [2283, 4], [2286, 3], [2289, 2], [2290, 3], [2291, 5], [2294, 2], [2297, 1], [2300, 5], [2303, 2], [2306, 1], [2309, 5], [2312, 2], [2315, 1], [2316, 2], [2317, 3], [2320, 2], [2323, 1], [2326, 5], [2329, 2], [2332, 1], [2335, 5], [2338, 2], [2341, 1], [2344, 5], [2347, 2], [2350, 1], [2351, 2], [2354, 1], [2357, 5], [2360, 2], [2363, 1], [2366, 5], [2367, 4], [2370, 3], [2371, 5], [2374, 2], [2377, 1], [2380, 5], [2381, 4], [2384, 3], [2387, 2], [2390, 1], [2391, 2], [2392, 3], [2395, 2], [2398, 1], [2401, 5], [2402, 4], [2403, 2], [2404, 3], [2407, 2], [2410, 1], [2411, 2], [2414, 1], [2417, 5], [2420, 2], [2423, 1], [2426, 5], [2429, 2], [2432, 1], [2435, 5], [2438, 2], [2441, 1], [2444, 5], [2447, 2], [2450, 1], [2451, 2], [2452, 3], [2455, 2], [2458, 1], [2461, 5], [2464, 2], [2467, 1], [2470, 5], [2473, 2], [2476, 1], [2479, 5], [2482, 2], [2485, 1], [2486, 2], [2489, 1], [2492, 5], [2493, 4], [2496, 3], [2499, 2], [2500, 3], [2501, 5], [2504, 2], [2507, 1], [2510, 5], [2513, 2], [2516, 1], [2519, 5], [2522, 2], [2525, 1], [2526, 2], [2527, 3], [2530, 2], [2533, 1], [2536, 5], [2539, 2], [2542, 1], [2545, 5], [2548, 2], [2551, 1], [2554, 5], [2557, 2], [2560, 1]

before the circuit broke down. The sequence has the format [output, state number/light]. When we analyzed the parts, we could see some circular objects looking similar but with different sizes.

Can we determine how the circuit worked and recreate a functioning one using human technology? A solution should at least describe the rule how the numbers of generated.

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  • $\begingroup$ Is the mechanical-puzzles tag appropriate? Its tag description reads: "Puzzles that are physically made and have some sort of mechanism for turning or moving them, such as Rubik's Cubes or trick boxes." $\endgroup$ – Aza Dec 11 '15 at 23:51
  • $\begingroup$ Perhaps the OP intends that the solution involves a number of cogs. $\endgroup$ – Gordon K Dec 12 '15 at 8:51
  • $\begingroup$ @Emrakul In some sense it does, but not directly. I removed the tag now. $\endgroup$ – Carl Löndahl Dec 12 '15 at 9:32
3
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The first thing to notice is that whenever a red arrow is followed, the output increases by 3, while a black arrow increases the output by 1.

The arrow to be followed is determined by the current value of the output, according to the following table:

$$\begin{array}\\ &\text{State}&\text{Ouput}&\text{Arrow}&\text{Goes to}\\ \hline &1&\text{multiple of 5}&\text{black}&2\\ &1&\text{not multiple of 5}&\text{red}&5\\ &2&\text{multiple of 3}&\text{black}&3\\ &2&\text{not multiple of 3}&\text{red}&1\\ &3&\text{multiple of 10}&\text{black}&5\\ &3&\text{not multiple of 10}&\text{red}&2\\ &4&\text{multiple of 2}&\text{black}&2\\ &4&\text{not multiple of 2}&\text{red}&3\\ &5&\text{multiple of 7}&\text{black}&4\\ &5&\text{not multiple of 7}&\text{red}&2\\ \end{array} $$

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  • $\begingroup$ Nice! What was your method? My idea was to make very long list so that you have to derive an efficient method to determine the pattern. $\endgroup$ – Carl Löndahl Dec 15 '15 at 6:33
  • $\begingroup$ @CarlLöndahl I put the list into Excel, and highlighted all the places where it increased by 1 instead of 3. After trying various calculations, and staring at the list for a long time, I noticed that most of the outputs where the increase is 1 were multiples of 5 (I guess due to the fact that "many roads" lead to state #2). After staring at it a while longer, I determined that whenever it was in state #2, the number was a multiple of 5. After that, it was pretty easy to figure out the patterns for the other states. $\endgroup$ – GentlePurpleRain Dec 15 '15 at 6:37
  • $\begingroup$ People tend to like Excel on this site :-) But yeah, this was basically the method I intended, looking at the constant increments. Again, good job! $\endgroup$ – Carl Löndahl Dec 15 '15 at 6:47
2
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I'm not sure if there's supposed to be more to the answer than this...

The machine moves between the states 1 through 5. Following a red arrow adds $3$ to the output. Following a black arrow adds $1$ to the output.

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  • 5
    $\begingroup$ In each state, there is a choice between red and black. Presumably that choice isn't arbitrary? Perhaps that is the "more to this answer"... $\endgroup$ – Dr Xorile Dec 11 '15 at 23:04
  • $\begingroup$ @DrXorile is correct. The choice is not arbitrary and it is deterministic. $\endgroup$ – Carl Löndahl Dec 12 '15 at 9:29
  • $\begingroup$ I've discovered a pattern, but I haven't been able to derive a rule from it yet. Starting at output 61, the states follow the following pattern: 5,2,1,5,4,3,2,1,2,3,2,1,5,4,2,3,2,1,2,1,5,2,1,5,2,1,5,2,1,5,2,1,2,3,2,1,5,2,1,5,2,1,5,2,1,2,1,5,4,3,2,3,5,2,1,5,2,1,5,2,1,2,3,2,1,5,2,1,5,2,1,5,2,1,2,1,5,2,1,5,4,3. After this the same sequence begins again. $\endgroup$ – GentlePurpleRain Dec 14 '15 at 22:24
  • $\begingroup$ Is it just me, or any anyone else see the <kbd>1</kbd> and <kbd>5</kbd> without mousing over? Pardon me if this should be on meta. $\endgroup$ – JTL Dec 15 '15 at 4:32
  • $\begingroup$ @JTL Yes, it looks the same to me. $\endgroup$ – GentlePurpleRain Dec 15 '15 at 6:30

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