collisions in hash functions are unavoidable. if you find one collision (messages A and B have the same hash), how easy is it to manufacture another collision (messages C and D have the same hash), or even an infinite collection of collisions?
can cryptographic hash functions be designed to prevent collisions from easily yielding more collisions?
on official diagrams, the MBTA Red Line is drawn to split into the Ashmont and Braintree lines south of JFK/UMass station. in reality, the lines have already split north of the station, and the station has separate platforms for each line. to catch the next northbound (inbound) train, you have to guess which platform to wait on. there is technology that announces which platform will have the next northbound train, but I don't know how reliable it is or how much advance warning it gives, in case stairs or elevator are slow for you.
but wait, it gets worse. even though the lines have split as of JFK/UMass, the tracks remain parallel and next to each other through the next station to the south, Savin Hill. only trains on the Ashmont line stop at Savin Hill. therefore, to get from North Quincy on the Braintree line to Savin Hill, you board the northbound train at North Quincy, watch from your train as it goes past but does not stop at Savin Hill, get off at JFK/UMass, walk up and over to the Ashmont platform, then take a southbound train back to Savin Hill.
who on earth thought all this was a good idea? I suspect it has something to do with communities on the Ashmont line being predominantly poor and African-American.
redraw the train diagram to reflect reality, as a way of drawing attention to the stupidity. JFK/UMass is effectively two stations with free transfer between them. both branches go through Savin Hill, but only the Ashmont line gets a white dot indicating the train stops there. only after Savin Hill do the lines go separate directions.
one of the few places that multiples of 10 exist in U.S. customary units is obscure units of area:
10 square chain = acre
10 acre = square furlong
however, this feels awkward because 10 congruent squares cannot be arranged into a larger square. it would have been nice if our counting system were based on a square number.
acre is a square with side length sqrt(10) ~= 3.162277660168379332 chain, or 0.3162277660168379332 furlong.
compare this with the metric system which does area in powers of 100: square meter, are, hectare, square kilometer, (megaare? megare?), (hectomegaare?), square megameter. (but previously on sqrt(10) elsewhere in the metric system.)
64 square furlong in a square mile, realizable as an 8x8 array of square furlongs. 100 square chain in a square furlong.
partition a square into 10 congruent pieces. 10 strips, or a 2x5 array of rectangles, work. are there any other ways?
an acre is a strip 1 chain by 10 chain (22 yard by 220 yard).
(a football field is (53 + 1/3) yard wide, or (2 + 14/33) chain. including end zones, it has area (1 + 117/363) = (1 + 39/121) acre. excluding end zones, (1 + 37/363) acre.)
partition a square into 10 regions of equal area, optimizing for compactness of the regions by some measure.
Arlington is the town where people "arl". Birmingham is the ham (old French for "village") where people "birm".
no one knows what those words mean, so invent something:
having lived in Arlington, I can confirm that people there actually do arl, a lot. I will never forget the looks on the faces of those gerbils.
prove that spherically symmetric redistributions of mass do not emit gravitational waves. (you are smarter than Einstein.)
"welcome aboard __________ Airlines, where are priorities are
obtaining a whole bunch of regular dice (d6) and rolling them simultaneously seems not very difficult. (Amazon currently sells sets of as many as 100 for under $20.) consider rolling until they all come up 6. doing such an exercise allows viscerally trying to experience very low probabilities. humans generally have difficulty understanding low probabilities. for a given low probability, you can map it to a number of dice.
below are the reciprocals of probabilities of N dice up to 100, interspersed with some orders of magnitude in base 10 and base 2. for example, all sixes on 2 dice has a probability of 1 in 36. 100 dice is approximately the probability of breaking 258-bit security by random chance.
is it practically difficult to roll many dice simultaneously? maybe they land leaned against each other, interfering with landing flat. beyond a certain number, it becomes difficult to verify between rolls that you haven't lost any.
coins are also easy to obtain, but when flipping many coins simultaneously, it's likely some will roll quite far on their edges before landing on a side. collisions among dice also might occasionally propel some of them a very long distance.
1 6 (6.0e0)
2 36 (3.6e1)
3 216 (2.2e2)
4 1,296 (1.3e3)
5 7,776 (7.8e3)
6 46,656 (4.7e4)
7 279,936 (2.8e5)
million
8 1,679,616 (1.7e6)
9 10,077,696 (1.0e7)
10 60,466,176 (6.0e7)
11 362,797,056 (3.6e8)
billion
12 2,176,782,336 (2.2e9)
13 13,060,694,016 (1.3e10)
14 78,364,164,096 (7.8e10)
15 470,184,984,576 (4.7e11)
trillion
16 2,821,109,907,456 (2.8e12)
17 16,926,659,444,736 (1.7e13)
18 101,559,956,668,416 (1.0e14)
19 609,359,740,010,496 (6.1e14)
quadrillion
20 3,656,158,440,062,976 (3.7e15)
21 21,936,950,640,377,856 (2.2e16)
22 131,621,703,842,267,136 (1.3e17)
23 789,730,223,053,602,816 (7.9e17)
24 4,738,381,338,321,616,896 (4.7e18)
64 bits
25 28,430,288,029,929,701,376 (2.8e19)
26 170,581,728,179,578,208,256 (1.7e20)
27 1,023,490,369,077,469,249,536 (1.0e21)
28 6,140,942,214,464,815,497,216 (6.1e21)
29 36,845,653,286,788,892,983,296 (3.7e22)
30 221,073,919,720,733,357,899,776 (2.2e23)
31 1,326,443,518,324,400,147,398,656 (1.3e24)
32 7,958,661,109,946,400,884,391,936 (8.0e24)
33 47,751,966,659,678,405,306,351,616 (4.8e25)
34 286,511,799,958,070,431,838,109,696 (2.9e26)
35 1,719,070,799,748,422,591,028,658,176 (1.7e27)
36 10,314,424,798,490,535,546,171,949,056 (1.0e28)
37 61,886,548,790,943,213,277,031,694,336 (6.2e28)
38 371,319,292,745,659,279,662,190,166,016 (3.7e29)
39 2,227,915,756,473,955,677,973,140,996,096 (2.2e30)
40 13,367,494,538,843,734,067,838,845,976,576 (1.3e31)
41 80,204,967,233,062,404,407,033,075,859,456 (8.0e31)
42 481,229,803,398,374,426,442,198,455,156,736 (4.8e32)
43 2,887,378,820,390,246,558,653,190,730,940,416 (2.9e33)
44 17,324,272,922,341,479,351,919,144,385,642,496 (1.7e34)
45 103,945,637,534,048,876,111,514,866,313,854,976 (1.0e35)
46 623,673,825,204,293,256,669,089,197,883,129,856 (6.2e35)
47 3,742,042,951,225,759,540,014,535,187,298,779,136 (3.7e36)
48 22,452,257,707,354,557,240,087,211,123,792,674,816 (2.2e37)
49 134,713,546,244,127,343,440,523,266,742,756,048,896 (1.3e38)
128 bits
50 808,281,277,464,764,060,643,139,600,456,536,293,376 (8.1e38)
51 4,849,687,664,788,584,363,858,837,602,739,217,760,256 (4.8e39)
52 29,098,125,988,731,506,183,153,025,616,435,306,561,536 (2.9e40)
53 174,588,755,932,389,037,098,918,153,698,611,839,369,216 (1.7e41)
54 1,047,532,535,594,334,222,593,508,922,191,671,036,215,296 (1.0e42)
55 6,285,195,213,566,005,335,561,053,533,150,026,217,291,776 (6.3e42)
56 37,711,171,281,396,032,013,366,321,198,900,157,303,750,656 (3.8e43)
57 226,267,027,688,376,192,080,197,927,193,400,943,822,503,936 (2.3e44)
58 1,357,602,166,130,257,152,481,187,563,160,405,662,935,023,616 (1.4e45)
59 8,145,612,996,781,542,914,887,125,378,962,433,977,610,141,696 (8.1e45)
60 48,873,677,980,689,257,489,322,752,273,774,603,865,660,850,176 (4.9e46)
61 293,242,067,884,135,544,935,936,513,642,647,623,193,965,101,056 (2.9e47)
62 1,759,452,407,304,813,269,615,619,081,855,885,739,163,790,606,336 (1.8e48)
63 10,556,714,443,828,879,617,693,714,491,135,314,434,982,743,638,016 (1.1e49)
64 63,340,286,662,973,277,706,162,286,946,811,886,609,896,461,828,096 (6.3e49)
65 380,041,719,977,839,666,236,973,721,680,871,319,659,378,770,968,576 (3.8e50)
66 2,280,250,319,867,037,997,421,842,330,085,227,917,956,272,625,811,456 (2.3e51)
67 13,681,501,919,202,227,984,531,053,980,511,367,507,737,635,754,868,736 (1.4e52)
68 82,089,011,515,213,367,907,186,323,883,068,205,046,425,814,529,212,416 (8.2e52)
69 492,534,069,091,280,207,443,117,943,298,409,230,278,554,887,175,274,496 (4.9e53)
70 2,955,204,414,547,681,244,658,707,659,790,455,381,671,329,323,051,646,976 (3.0e54)
71 17,731,226,487,286,087,467,952,245,958,742,732,290,027,975,938,309,881,856 (1.8e55)
72 106,387,358,923,716,524,807,713,475,752,456,393,740,167,855,629,859,291,136 (1.1e56)
73 638,324,153,542,299,148,846,280,854,514,738,362,441,007,133,779,155,746,816 (6.4e56)
74 3,829,944,921,253,794,893,077,685,127,088,430,174,646,042,802,674,934,480,896 (3.8e57)
192 bits
75 22,979,669,527,522,769,358,466,110,762,530,581,047,876,256,816,049,606,885,376 (2.3e58)
76 137,878,017,165,136,616,150,796,664,575,183,486,287,257,540,896,297,641,312,256 (1.4e59)
77 827,268,102,990,819,696,904,779,987,451,100,917,723,545,245,377,785,847,873,536 (8.3e59)
78 4,963,608,617,944,918,181,428,679,924,706,605,506,341,271,472,266,715,087,241,216 (5.0e60)
79 29,781,651,707,669,509,088,572,079,548,239,633,038,047,628,833,600,290,523,447,296 (3.0e61)
80 178,689,910,246,017,054,531,432,477,289,437,798,228,285,773,001,601,743,140,683,776 (1.8e62)
81 1,072,139,461,476,102,327,188,594,863,736,626,789,369,714,638,009,610,458,844,102,656 (1.1e63)
82 6,432,836,768,856,613,963,131,569,182,419,760,736,218,287,828,057,662,753,064,615,936 (6.4e63)
83 38,597,020,613,139,683,778,789,415,094,518,564,417,309,726,968,345,976,518,387,695,616 (3.9e64)
84 231,582,123,678,838,102,672,736,490,567,111,386,503,858,361,810,075,859,110,326,173,696 (2.3e65)
85 1,389,492,742,073,028,616,036,418,943,402,668,319,023,150,170,860,455,154,661,957,042,176 (1.4e66)
86 8,336,956,452,438,171,696,218,513,660,416,009,914,138,901,025,162,730,927,971,742,253,056 (8.3e66)
87 50,021,738,714,629,030,177,311,081,962,496,059,484,833,406,150,976,385,567,830,453,518,336 (5.0e67)
88 300,130,432,287,774,181,063,866,491,774,976,356,909,000,436,905,858,313,406,982,721,110,016 (3.0e68)
89 1,800,782,593,726,645,086,383,198,950,649,858,141,454,002,621,435,149,880,441,896,326,660,096 (1.8e69)
90 10,804,695,562,359,870,518,299,193,703,899,148,848,724,015,728,610,899,282,651,377,959,960,576 (1.1e70)
91 64,828,173,374,159,223,109,795,162,223,394,893,092,344,094,371,665,395,695,908,267,759,763,456 (6.5e70)
92 388,969,040,244,955,338,658,770,973,340,369,358,554,064,566,229,992,374,175,449,606,558,580,736 (3.9e71)
93 2,333,814,241,469,732,031,952,625,840,042,216,151,324,387,397,379,954,245,052,697,639,351,484,416 (2.3e72)
94 14,002,885,448,818,392,191,715,755,040,253,296,907,946,324,384,279,725,470,316,185,836,108,906,496 (1.4e73)
95 84,017,312,692,910,353,150,294,530,241,519,781,447,677,946,305,678,352,821,897,115,016,653,438,976 (8.4e73)
96 504,103,876,157,462,118,901,767,181,449,118,688,686,067,677,834,070,116,931,382,690,099,920,633,856 (5.0e74)
97 3,024,623,256,944,772,713,410,603,088,694,712,132,116,406,067,004,420,701,588,296,140,599,523,803,136 (3.0e75)
98 18,147,739,541,668,636,280,463,618,532,168,272,792,698,436,402,026,524,209,529,776,843,597,142,818,816 (1.8e76)
99 108,886,437,250,011,817,682,781,711,193,009,636,756,190,618,412,159,145,257,178,661,061,582,856,912,896 (1.1e77)
256 bits
100 653,318,623,500,070,906,096,690,267,158,057,820,537,143,710,472,954,871,543,071,966,369,497,141,477,376 (6.5e77)
it is deeply ironic that one of the fears antivaxxers have about COVID-19 vaccination is the fear that mRNA vaccines will rewrite your DNA. it is ironic because COVID-19 the disease will absolutely definitely rewrite your DNA. because that's what viruses do: they rewrite your DNA, reprogramming your own cells to become virus factories. viruses are very good at rewriting your DNA.
if it turns out that children and future descendants of people infected with COVID-19 have a higher rate of birth defects or autism or homosexuality or whatever, no biologist will be very surprised, because that's what viruses do: they rewrite your DNA. most of the time, the COVID-19 virus (SARS-CoV-2) infects and rewrites the DNA of cells in your respiratory system, but everything is connected inside your body, so it is completely plausible that the virus might manage to infect a gamete (sperm or egg cell) and thereby rewrite the DNA of your yet-to-be-conceived children and all your future descendants.
the human genome is absolutely littered with the DNA of viruses which successfully rewrote the DNA of some human ancestor. most of those virus sequences have been rendered inert by evolution (virus rewrites the human DNA of something critical, baby is stillborn, that virus sequence then fails to propagate through human DNA), but it is an open research question how much virus DNA is still active, how much of what makes us human is actually the result of ancestral viruses which rewrote our DNA. previously.
if you want to protect the sanctity of your DNA, get vaccinated.
(counterargument: presume that the government (secretly puppeteered of course by the Illuminati, Rothschilds, Freemasons, Lizard People, etc.) and COVID-19 are both equally competent at rewriting your DNA if they are allowed to introduce a foreign substance into your body. in choosing whether to get vaccinated or let the disease put virus into you, you are trying to choose the lesser evil. the government has both a documented history of being evil and plainly visible incentives to continue to be evil in order to maintain power.)
each composite number below consists of repetitions of the digit string 1234567890 except the final few digits. the smaller factors are large enough that quadratic sieve is probably more efficient than elliptic curve method (ECM).
previously similar for base 2, assuming you have an expression parser.
40 digits: 1234567890123456789012345678901234567759 = 5284744045008013 * 233609779321220631386443
50 digits: 12345678901234567890123456789012345678901234567633 = 92279162895906968524229 * 133786203881810018387538077
60 digits: 123456789012345678901234567890123456789012345678901234567831 = 87988560978576315942371700757 * 1403100444413510291678340665083
70 digits: 1234567890123456789012345678901234567890123456789012345678901234566713 = 61208527479308458227554760703 * 20169867516267279020392646921656369545671
80 digits: 12345678901234567890123456789012345678901234567890123456789012345678901234567751 = 575099128882847693294536030177708942207 * 21467045038333698701383939676398791777593
90 digits: 123456789012345678901234567890123456789012345678901234567890123456789012345678901234567597 = 40166540925345636983244707230202848993 * 3073622626399545288925403158158369054873454157207629
model an incompressible liquid under the influence of its own gravity. the completely static solution is a sphere, and there is probably a rotating hydrostatic solution the shape of an oblate ellipsoid. exactly which ellipsoidoid (because might not exactly be an ellipsoid)?
what if general relativity?
if the fluid is allowed to move, then things likely become insane due to Navier-Stokes. turbulence might cause droplets to break off and be ejected faster than escape velocity.
the plethora of packages whose names end in "t64" recently added to Debian Sid (Unstable) change the time_t data type from 32 to 64 bits, pushing off the Year 2038 Problem to the year 292,277,026,596. (previously, 835 bits should be enough for everyone.)
some package examples seen starting 2024-02-28:
libapt-pkg6.0t64
libdb5.3t64
libpam0t64
libssl3t64
