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1047: Approximations

2025-02-15T22:08:59Z

172.70.210.235: /* Proof */

{{comic
| number = 1047
| date = April 25, 2012
| title = Approximations
| before = [[#Explanation|↓ Skip to explanation ↓]]
| image = approximations.png
| titletext = Two tips: 1) 8675309 is not just prime, it's a twin prime, and 2) if you ever find yourself raising log(anything)^e or taking the pi-th root of anything, set down the marker and back away from the whiteboard; something has gone horribly wrong.
}}

==Explanation==

This comic lists some approximations for numbers, most of them mathematical and physical constants, but some of them jokes and cultural references.

Approximations like these are sometimes used as {{w|mnemonic}}s by mathematicians and physicists, though most of Randall's approximations are too convoluted to be useful as mnemonics. Perhaps the best known mnemonic approximation (though not used here by Randall) is that "π is approximately equal to 22/7". Randall does mention (and mock) the common mnemonic among physicists that the {{w|fine structure constant}} is approximately 1/137. Although Randall gives approximations for the number of seconds in a year, he does not mention the common physicists' mnemonic that it is "π × 107", though he later added a statement to the top of the comic page addressing this point.

At the bottom of the comic are expressions involving {{w|transcendental numbers}} (namely π and e) that are tantalizingly close to being exactly true but are not (indeed, they cannot be, due to the nature of transcendental numbers). Such near-equations were previously discussed in [[217: e to the pi Minus pi]]. One of the entries, though, is a "red herring" that is exactly true.

Randall says he compiled this table through "a mix of trial-and-error, ''{{w|Mathematica}}'', and Robert Munafo's [http://mrob.com/pub/ries/ Ries] tool." "Ries" is a "{{w|Closed-form expression#Conversion from numerical forms|reverse calculator}}" that forms equations matching a given number.

The {{w|world population}} estimate for 2025 is still somewhat accurate. The estimate is 8.0 billion, and the population listed at the website census.gov is roughly the same. The current value can be found here: [https://www.census.gov/popclock/ United States Census Bureau - U.S. and World Population Clock]. Nevertheless, there are other numbers listed by different sources.

The first part of the title text notes that "Jenny's constant," which is actually a telephone number referenced in Tommy Tutone's 1982 song {{w|867-5309/Jenny}}, is not only prime but a {{w|twin prime}} because 8675311 is also a prime. Twin primes have always been a subject of interest, because they are comparatively rare, and because it is not yet known whether there are infinitely many of them. Twin primes were also referenced in [[1310: Goldbach Conjectures]].

The second part of the title text makes fun of the unusual mathematical operations contained in the comic. {{w|Pi|π}} is a useful number in many contexts, but it doesn't usually occur anywhere in an exponent. Even when it does, such as with complex numbers, taking the πth root is rarely helpful. A rare exception is an [http://gosper.org/4%5E1%C3%B7%CF%80.png identity] for the pi-th root of 4 discovered by Bill Gosper. Similarly, {{w|e (mathematical constant)|e}} typically appears in the base of a power (forming the {{w|exponential function}}), not in the exponent. (This is later referenced in [http://what-if.xkcd.com/73/ Lethal Neutrinos]).

In [[217: e to the pi Minus pi]] and [[3023: The Maritime Approximation]] Randall gives other approximations based on numerical coincidences.

===Equations===

{| class="wikitable"
|-
!align="center"|Thing to be approximated:
!align="center"|Formula proposed
!align="center"|Resulting approximate value
!align="center"|Correct value
!align="center"|Discussion
!align="center"|Error
|-
|align="center"|One {{w|light year}} (meters)
|align="center"|998
|align="center"|9,227,446,944,279,201
|align="center"|9,460,730,472,580,800 (exact)
|align="left"|Based on 365.25 days per year (see below). 998 and 698 are [[487: Numerical Sex Positions|sexual references]].
|align="center"|2.3328353 × 1014
|-
|align="center"|Earth's surface (m2)
|align="center"|698
|align="center"|513,798,374,428,641
|align="center"|5.10072 × 1014
|align="left"|998 and 698 are [[487: Numerical Sex Positions|sexual references]].
|align="center"|3.7263744 × 1012
|-
|align="center"|Oceans' volume (m3)
|align="center"|919
|align="center"|1,350,851,717,672,992,089
|align="center"|1.332 × 1018
|align="left"|
|-
|align="center"|Seconds in a year
|align="center"|754
|align="center"|31,640,625
|align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
|align="left"|After this comic was released [[Randall]] got many responses by viewers. So he did add this statement to the top of the comic page:
"Lots of emails mention the physicist favorite, 1 year = pi × 107 seconds. 754 is a hair more accurate, but it's hard to top 3,141,592's elegance." π × 107 is nearly equal to 31,415,926.536, and 754 is exactly 31,640,625. Randall's elegance belongs to the number π, but it should be multiplied by the factor of ten.

Using the traditional definitions that a second is 1/60 of a minute, a minute is 1/60 of an hour, and an hour is 1/24 of a day, a 365-day common year is exactly 31,536,000 seconds (the "''Rent'' method" approximation) and the 366-day leap year is 31,622,400 seconds. Until the calendar was reformed by Pope Gregory, there was one leap year in every four years, making the average year 365.25 days, or 31,557,600 seconds. On the current calendar system, there are only 97 leap years in every 400 years, making the average year 365.2425 days, or 31,556,952 seconds. In technical usage, a "second" is now defined based on physical constants, even though the length of a day varies inversely with the changing angular velocity of the earth. To keep the official time synchronized with the rotation of the earth, a "leap second" is occasionally added, resulting in a slightly longer year.
|-
|align="center"|Seconds in a year (''Rent'' method)
|align="center"|525,600 × 60
|align="center"|31,536,000
|align="center"|31,557,600 (Julian calendar), 31,556,952 (Gregorian calendar)
|align="left"|"''Rent'' Method" refers to the song "{{w|Seasons of Love}}" from the musical ''{{w|Rent (musical)|Rent}}''. The song asks, "How do you measure a year?" One line says "525,600 minutes" while most of the rest of the song suggests the best way to measure a year is moments shared with a loved one. This method for remembering how many seconds are in a year was also referenced in [https://what-if.xkcd.com/23/ What If? 23].
|-
|align="center"|Age of the universe (seconds)
|align="center"|1515
|align="center"|437,893,890,380,859,375
|align="center"|(4.354 ± 0.012) × 1017 (best estimate; exact value unknown)
|align="left"|This one will slowly get more accurate as the universe ages.
|-
|align="center"|Planck's constant
|align="center"|<math>\frac {1} {30^{\pi^e}}</math>
|align="center"|6.6849901410 × 10−34
|align="center"|6.62606957 × 10−34
|align="left"|Informally, the {{w|Planck constant}} is the smallest action possible in quantum mechanics.
|-
|align="center"|Fine structure constant
|align="center"|<math>\frac{1}{140}</math>
|align="center"|0.00714285
|align="center"|0.0072973525664 (accepted value as of 2014), close to 1/137
|align="left"|The {{w|fine structure constant}} indicates the strength of electromagnetism. It is unitless and around 0.007297, close to 1/137. The joke here is that Randall chose to write 140 as the denominator, when 137 is much closer to reality and just as many digits (although 137 is a less "round" number than 140, and Randall writes in the table that he's "had enough" of it). At one point the fine structure constant was believed to be exactly the reciprocal of 137, and many people have tried to find a simple formula explaining this (with a pinch of {{w|numerology}} thrown in at times), including the infamous {{w|Arthur Eddington|Sir Arthur "Adding-One" Eddington}} who argued very strenuously that the fine structure constant "should" be 1/136 when that was what the best measurements suggested, and then argued just as strenuously for 1/137 a few years later as measurements improved.
|-
|align="center"|Fundamental charge
|align="center"|<math>\frac {3} {14 \pi^{\pi^\pi}}</math>
|align="center"|1.59895121062716 × 10−19
|align="center"|1.602176565 × 10−19
|align="left"|This is the charge of the proton, symbolized ''e'' for electron (whose charge is actually −''e''. You can blame Benjamin Franklin [[567|for that]].)
|-
|align="center"|Telephone number for the {{w|White House}} switchboard
|align="center"|<math>\frac {1} {e^ {\sqrt[\pi] {1 + \sqrt[e-1] 8}} }</math>
|align="center"|0.2024561414932
|align="center"|202-456-1414
|align="left"|
|-
|align="center"|Jenny's constant
|align="center"|<math>\left( 7^ {\frac{e}{1} - \frac{1}{e}} - 9 \right) \pi^2</math>
|align="center"|867.5309019
|align="center"|867-5309
|align="left"|A telephone number referenced in {{w|Tommy Tutone}}'s 1982 song {{w|867-5309/Jenny}}. As mentioned in the title text, the number is not only prime but a {{w|twin prime}} because 8675311 is also a prime.
|-
|align="center"|World population estimate (billions)
|align="center"|Equivalent to <math>6 + \frac {\frac34 y + \frac14 (y \operatorname{mod} 4) - 1499} {10}</math>
|align="center"|2005 — 6.5 
2006 — 6.6 
2007 — 6.7 
2008 — 6.7 
2009 — 6.8 
2010 — 6.9 
2011 — 7.0 
2012 — 7.0 
2013 — 7.1 
2014 — 7.2 
2015 — 7.3 
2016 — 7.3 
2017 — 7.4 
2018 — 7.5 
2019 — 7.6 
2020 — 7.6 
2021 — 7.7 
2022 — 7.8 
2023 — 7.9 
2024 — 7.9 
2025 — 8.0 
2026 — 8.1 
2027 — 8.2 
2028 — 8.2 
2029 — 8.3 
2030 — 8.4 
2031 — 8.5 
2032 — 8.5 
2033 — 8.6 
2034 — 8.7 
2035 — 8.8 
|align="center"|
|align="left"|Grows by 75 million every year on average (100 million every year, except for a pause every leap-year). As of 2024, still a bit too small. (7.8B)
|-
|align="center"|U.S. population estimate (millions)
|align="center"|Equivalent to <math>310 + 3(y - 2010)</math>
|align="center"|2000 — 280 
2001 — 283 
2002 — 286 
2003 — 289 
2004 — 292 
2005 — 295 
2006 — 298 
2007 — 301 
2008 — 304 
2009 — 307 
2010 — 310 
2011 — 313 
2012 — 316 
2013 — 319 
2014 — 322 
2015 — 325 
2016 — 328 
2017 — 331 
2018 — 334 
2019 — 337 
2020 — 340 
2021 — 343 
2022 — 346 
2023 — 349 
2024 — 352 
2025 — 355 
2026 — 358 
2027 — 361 
2028 — 364 
2029 — 367 
2030 — 370 
2031 — 373 
2032 — 376 
2033 — 379 
2034 — 382 
2035 — 385 
|align="center"|
|align="left"|Grows by 3 million each year. As of 2024 the actual number is ~10 million smaller.
|-
|align="center"|Electron rest energy (joules)
|align="center"|<math>\frac {e} {7^{16}}</math>
|align="center"|8.17948276564429 × 10−14
|align="center"|8.18710438 × 10−14
|align="left"|
|-
|align="center"|Light year (miles)
|align="center"|242.42
|align="center"|5,884,267,614,436.97
|align="center"|5,878,625,373,183.61 = 9,460,730,472,580,800 (meters in a light-year, by definition) / 1609.344 (meters in a mile)
|align="left"|{{w|42 (number)|42}} is, according to {{w|Douglas Adams}}' ''{{w|The Hitchhiker's Guide to the Galaxy}}'', the answer to the Ultimate Question of Life, the Universe, and Everything.
|-
|align="center"|<math>\sin\left(60^\circ\right) = \frac {\sqrt 3} {2}</math>
|align="center"|<math>\frac{e}{\pi}</math>
|align="center"|0.8652559794
|align="center"|0.8660254038
|align="left"|
|-
|align="center"|<math>\sqrt 3</math>
|align="center"|<math>\frac{2e}{\pi}</math>
|align="center"|1.7305119589
|align="center"|1.7320508076
|align="left"|Same as the above
|-
|align="center"|γ (Euler's gamma constant)
|align="center"|<math>\frac {1} {\sqrt 3}</math>
|align="center"|0.5773502692
|align="center"|0.5772156649
|align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
|-
|align="center"|Feet in a meter
|align="center"|<math>\frac {5} {\sqrt[e]\pi}</math>
|align="center"|3.2815481951
|align="center"|3.280839895
|align="left"|Exactly 1/0.3048, as the {{w|international foot}} is defined as 0.3048 meters.
|-
|align="center"|<math>\sqrt 5</math>
|align="center"|<math>\frac{2}{e} + \frac32</math>
|align="center"|2.2357588823
|align="center"|2.2360679775
|align="left"|
|-
|align="center"|Avogadro's number
|align="center"|<math>69^{\pi^\sqrt{5}}</math>
|align="center"|6.02191201246329 × 1023
|align="center"|6.02214129 × 1023
|align="left"|Also called a mole for shorthand, {{w|Avogadro's number}} is (roughly) the number of individual atoms in 12 grams of pure carbon. Used in basically every application of chemistry. In 2019 the constant was redefined to 6.02214076 × 1023, making the Approximation slightly more correct.
|-
|align="center"|Gravitational constant ''G''
|align="center"|<math>\frac {1} {e ^ {(\pi-1)^{(\pi+1)}}}</math>
|align="center"|6.6736110685 × 10−11
|align="center"|6.67385 × 10−11
|align="left"|The universal {{w|gravitational constant}} G is equal to ''Fr''2/''Mm'', where ''F'' is the gravitational force between two objects, ''r'' is the distance between them, and ''M'' and ''m'' are their masses.
|-
|align="center"|''R'' (gas constant)
|align="center"|<math>(e + 1) \sqrt 5</math>
|align="center"|8.3143309279
|align="center"|8.3144622
|align="left"|The {{w|gas constant}} relates energy to temperature in physics, as well as a gas's volume, pressure, temperature and {{w|mole (unit)|molar amount}} (hence the name).
|-
|align="center"|Proton–electron mass ratio
|align="center"|<math>6 \pi^5</math>
|align="center"|1836.1181087117
|align="center"|1836.15267246
|align="left"| The {{w|proton-to-electron mass ratio}} is the ratio between the rest mass of the proton divided by the rest mass of the electron.
|-
|align="center"|Liters in a {{w|gallon}}
|align="center"|<math>3 + \frac{\pi}{4}</math>
|align="center"|3.7853981634
|align="center"|3.785411784 (exact)
|align="left"|A U.S. liquid gallon is defined by law as 231 cubic inches. The British imperial gallon would be about 20% larger (but the litre is the same thing as the US liter).
|-
|align="center"|''g''0 or ''g''n
|align="center"|6 + ln(45)
|align="center"|9.8066624898
|align="center"|9.80665
|align="left"|Standard gravity, or standard acceleration due to free fall is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2, which is exactly 35.30394 km/h/s (about 32.174 ft/s2, or 21.937 mph/s). This value was established by the 3rd CGPM (1901, CR 70) and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from rotation of the Earth (but which is small enough to be neglected for most purposes); the total (the apparent gravity) is about 0.5 percent greater at the poles than at the equator.

Randall used a letter ''g'' without a suffix, which can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth.
|-
|align="center"|Proton–electron mass ratio
|align="center"|<math>\frac {e^8 - 10} {\phi}</math>
|align="center"|1836.1530151398
|align="center"|1836.15267246
|align="left"|φ is the {{w|golden ratio}}, or <math>\textstyle{ \frac{1+\sqrt 5}{2} }</math>. It has many interesting geometrical properties.
|-
|align="center"|Ruby laser wavelength (meters)
|align="center"|<math>\frac{1}{1200^2}</math>
|align="center"|6.9444 × 10−7
|align="center"|~6.943 × 10−7
|align="left"|The {{w|ruby laser}} wavelength varies because "ruby" is not clearly defined.
|-
|align="center"|Mean Earth radius (meters)
|align="center"|<math>5^8 6e</math>
|align="center"|6,370,973.035
|align="center"|6,371,008.7 (IUGG definition)
|align="left"|The {{w|Earth radius#mean radii|mean earth radius}} varies because there is not one single way to make a sphere out of the earth. Randall's value lies within the actual variation of Earth's radius. The International Union of Geodesy and Geophysics (IUGG) defines the mean radius as 2/3 of the equatorial radius (6,378,137.0 m) plus 1/3 of the polar radius (6,356,752.3 m).
|-
|align="center"|<math>\sqrt 2</math>
|align="center"|<math>\frac35 + \frac{\pi}{7-\pi}</math>
|align="center"|1.4142200581
|align="center"|1.4142135624
|align="left"|There are recurring math jokes along the lines of, "<math>\textstyle{ \frac35 + \frac{\pi}{7-\pi} - \sqrt{2} = 0}</math>, but your calculator is probably not good enough to compute this correctly". See also [[217: e to the pi Minus pi]].
|-
|align="center"|<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>
|align="center"|<math>\frac12</math>
|align="center"|0.5
|align="center"|0.5 (exact)
|align="left"|This is the exactly correct equation referred to in the note, "Pro tip – Not all of these are wrong", as shown below and also [http://math.stackexchange.com/questions/140388/how-can-one-prove-cos-pi-7-cos3-pi-7-cos5-pi-7-1-2 here]. If you're still confused, the functions use {{w|radians}}, not {{w|degrees (angle)|degrees}}: when an angular measure does not specify units, radians are the assumed default.
|-
|align="center"|γ (Euler's gamma constant)
|align="center"|<math>\frac{e}{3^4} + \frac{e}{5}</math>
|align="center"|0.5772154006
|align="center"|0.5772156649
|align="left"|The {{w|Euler–Mascheroni constant}} (denoted γ) is a mysterious number describing the relationship between the {{w|Harmonic series (mathematics)|harmonic series}} and the {{w|natural logarithm}}.
|-
|align="center"|<math>\sqrt 5</math>
|align="center"|<math>\frac {13+4\pi} {24-4\pi}</math>
|align="center"|2.2360678094
|align="center"|2.2360679775
|align="left"|
|-
|align="center"|<math>\sum_{n=1}^{\infty} \frac{1}{n^n}</math>
|align="center"|<math>\ln(3)^e</math>
|align="center"|1.2912987577
|align="center"|1.2912859971
|align="left"|
|}

===Proof===

One of the "approximations" actually is precisely correct: <math>\textstyle{ \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} = \frac12 }</math>. Here is a proof:

:<math>\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7}</math>

Multiplying by 1 (or by a nonzero number divided by itself) leaves the equation unchanged:

:<math>= \left( \cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} \right) \frac{2 \sin\frac{\pi}{7}}{2 \sin\frac{\pi}{7}}</math>

The <math>\textstyle{ 2 \sin\frac{\pi}{7} }</math> on the top of the fraction is multiplied through the original equation:

:<math>= \frac {2 \cos \frac{\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{3\pi}{7} \sin\frac{\pi}{7} + 2 \cos \frac{5\pi}{7} \sin\frac{\pi}{7}} {2 \sin\frac{\pi}{7}}</math>

Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin B = \sin (A+B) - \sin(A-B)}</math> on the second and third terms in the numerator:

:<math>\begin{align}
&= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \left(\frac{3\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{3\pi}{7} - \frac{\pi}{7}\right) \right] + \left[\sin \left(\frac{5\pi}{7} + \frac{\pi}{7}\right) - \sin \left(\frac{5\pi}{7} - \frac{\pi}{7}\right) \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {2 \cos \frac{\pi}{7} \sin \frac{\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}}
\end{align}</math>

Use the trigonometric identity <math>\textstyle{ 2 \cos A \sin A = \sin 2A }</math> on the first term in the numerator:

:<math>\begin{align}
&= \frac {\sin \frac{2\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{2\pi}{7} \right] + \left[\sin \frac{6\pi}{7} - \sin \frac{4\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {\sin \frac{6\pi}{7} + \left[\sin \frac{4\pi}{7} - \sin \frac{4\pi}{7} \right] + \left[\sin \frac{2\pi}{7} - \sin \frac{2\pi}{7} \right]} {2 \sin\frac{\pi}{7}} \\
&= \frac {\sin \frac{6\pi}{7} } {2 \sin\frac{\pi}{7}}
\end{align}</math>

Noting that <math>\textstyle{\frac{6\pi}{7} + \frac{\pi}{7} = \pi}</math> and that the sines of supplementary angles (angles that sum to π) are equal:

:<math>\begin{align}
&= \frac {\sin \frac{\pi}{7} } {2 \sin\frac{\pi}{7}} \\
&= \frac12 \quad \quad \quad \text{Q.E.D.}
\end{align}</math>

To better see why the equation is true, it is better to go to the complex plane. cos(2k pi/7)  is the real part of the k-th 7-th root of unity, exp(2 k i pi/7). The seven 7-th roots of unity (for 0 <= k <= 6) sum up to zero, hence so do their real parts:


:0 = cos(0 pi/7) + cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) + cos(8 pi/7) + cos(10 pi/7) + cos(12 pi/7)

But one of these roots is just 1, and all other root go by pairs of conjugate roots, which have the same real part (alternatively, consider that cos(x) = cos(2 pi - x)):


:0 = 1 + 2 (cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7))

Hence


:cos(2 pi/7) + cos(4 pi/7) + cos(6 pi/7) = - 1/2

which, because cos(x) = -cos(pi - x), can be rewritten as


:cos(5 pi/7) + cos(3 pi/7) + cos(pi/7) = 1/2

Q.E.D.

==Transcript==
:'''A table of slightly wrong equations and identities useful for approximations and/or trolling teachers.'''
:(Found using a mix of trial-and-error, ''Mathematica'', and Robert Munafo's ''Ries'' tool.)
: All units are SI MKS unless otherwise noted.

:{| class="wikitable"
|-
|colspan="2" align="center" | Relation:
|align="center" | Accurate to within:
|-
|align="center" | One light-year(m)
|align="center" | 998
|align="center" | one part in 40
|-
|align="center" | Earth Surface(m2)
|align="center" | 698
|align="center" | one part in 130
|-
|align="center" | Oceans' volume(m3)
|align="center" | 919
|align="center" | one part in 70
|-
|align="center" | Seconds in a year
|align="center" | 754
|align="center" | one part in 400
|-
|align="center" | Seconds in a year (''Rent'' method)
|align="center" | 525,600 x 60
|align="center" | one part in 1400
|-
|align="center" | Age of the universe (seconds)
|align="center" | 1515
|align="center" | one part in 70
|-
|align="center" | Planck's constant
|align="center" | 1/(30πe)
|align="center" | one part in 110
|-
|align="center" | Fine structure constant
|align="center" | 1/140
|align="center" | [I've had enough of this 137 crap]
|-
|align="center" | Fundamental charge
|align="center" | 3/(14 * πππ)
|align="center" | one part in 500
|-
|align="center"|White House Switchboard
|colspan="2" align="center"|1 / (eπ√(1 + (e-1)√8))
|-
|align="center"|Jenny's Constant
|colspan="2" align="center"|(7(e/1 - 1/e) - 9) * π2
|-
|colspan="3" align="center"|Intermission: World Population Estimate which should stay current for a decade or two: 
Take the last two digits of the current year

Example: 20[14]

Subtract the number of leap years since hurricane Katrina

Example: 14 (minus 2008 and 2012) is 12

Add a decimal point

Example: 1.2

Add 6

Example: 6 + 1.2

7.2 = World population in billions.

Version for US population:

Example: 20[14]

Subtract 10

Example: 4

Multiply by 3

Example: 12

Add 10

Example: 3[22] million
|-
|align="center"|Electron rest energy
|align="center"|e/716 J
|align="center"|one part in 1000
|-
|align="center"|Light-year(miles)
|align="center"|2(42.42)
|align="center"|one part in 1000
|-
|colspan="2" align="center"|sin(60°) = √3/2 = e/π
|align="center"|one part in 1000
|-
|colspan="2" align="center"|√3 = 2e/π
|align="center"|one part in 1000
|-
|align="center"|γ(Euler's gamma constant)
|align="center"|1/√3
|align="center"|one part in 4000
|-
|align="center"|Feet in a meter
|align="center"|5/(e√π)
|align="center"|one part in 4000
|-
|colspan="2" align="center"|√5 = 2/e + 3/2
|align="center"|one part in 7000
|-
|align="center"|Avogadro's number
|align="center"|69π√5
|align="center"|one part in 25,000
|-
|align="center"|Gravitational constant G
|align="center"|1 / e(π - 1)(π + 1)
|align="center"|one part in 25,000
|-
|align="center"|R (gas constant)
|align="center"|(e+1) √5
|align="center"|one part in 50,000
|-
|align="center"|Proton-electron mass ratio
|align="center"|6*π5
|align="center"|one part in 50,000
|-
|align="center"|Liters in a gallon
|align="center"|3 + π/4
|align="center"|one part in 500,000
|-
|align="center"|g
|align="center"|6 + ln(45)
|align="center"|one part in 750,000
|-
|align="center"|Proton-electron mass ratio
|align="center"|(e8 - 10) / ϕ
|align="center"|one part in 5,000,000
|-
|align="center"|Ruby laser wavelength
|align="center"|1 / (12002)
|align="center"|[within actual variation]
|-
|align="center"|Mean Earth Radius
|align="center"|(58)*6e
|align="center"|[within actual variation]
|-
|colspan="3" align="center"|Protip - not all of these are wrong:
|-
|colspan="2" align="center"|√2 = 3/5 + π/(7-π)
|align="center"|cos(π/7) + cos(3π/7) + cos(5π/7) = 1/2
|-
|align="center"|γ(Euler's gamma constant) = e/34 + e/5
|align="center"|√5 = (13 + 4π) / (24 - 4π)
|align="center"|Σ 1/nn = ln(3)e
|}

{{comic discussion}}
[[Category:Charts]]
[[Category:Math]]
[[Category:Physics]]
[[Category:Protip]]
[[Category:Large drawings]]

3045: AlphaMove

2025-02-04T08:38:34Z

172.70.210.235: /* Explanation */ Elaborated on the pitfalls of attempting to account for the opponent using AlphaMove

{{comic
| number = 3045
| date = January 31, 2025
| title = AlphaMove
| image = alphamove_2x.png
| imagesize = 500x526px
| noexpand = true
| titletext = It struggles a little with complex positions, like when there are an even number of moves and it has to round down, but when run against itself it's capable of finding some novelties. At one point I saw six knights on the board at once; Stockfish rarely exceeds four.
}}

==Explanation==
{{incomplete|Created by THE SEVENTH KNIGHT, WAITING IN ANTICIPATION FOR THE BETA RELEASE. Do NOT delete this tag too soon.}}
This comic shows a new {{w|chess engine}}, presumably created by [[Randall]], which takes a list of all legal moves (in {{w|Algebraic notation (chess)|algebraic notation}}) in alphabetical order and chooses the median.

A playable implementation of this game can be found here: [https://enn-nafnlaus.github.io/AlphaMove/alphamove.html AlphaMove].

Algebraic notation begins with a symbol for which piece is being moved, which is always the first letter of the piece name except for knights (N) and pawns (nothing). This is then followed by the square that the piece is being moved to. (Rc4 would indicate a move that moved a rook to c4.) Other symbols include a lowercase x before the destination, indicating that the move is a capture; a plus sign (+) after the destination, indicating that the move places the opposing king in check; and a hash sign (#) after the destination, indicating that the move places the opposing king in checkmate, thus winning the game. There are also O-O and O-O-O notations, which indicate that a player is castling kingside or queenside, respectively.

In practice, this algorithm runs into a few issues. As seen in the comic, the algorithm rarely moves bishops and rooks due to their relative lack of moves in the early game, and their tendency to inhabit the edges of any list when they do have sortable moves. Among basic moves, only pawns destined to move in the first two files of a board can ever sort higher than bishops, and nothing other than another rook can be closer to the far end than a rook. The algorithm favors knight and king moves, with entries starting with the most alphabetically middling "K" and "N" list entries, and (to a lesser extent) pawns destined to move up the right side of the board, the "h"-file pawn generally having the greatest statistical chance. Castling moves would also occur near the middle of the list, as they are denoted with letter 'O' characters as O-O or O-O-O. The shortcomings of AlphaMove are instantly apparent from looking at the game board presented in the comic; roughly ten moves into the game, White has lost three pawns, captured nothing, and advanced their king into the open rather than developing any pieces, while Black (presumably being played by a 'stronger' conventional chess engine) has taken control of the center with a knight and two pawns, developed a bishop, and advanced their queen to f2.

The ''actual'' middle of the list might vary away from the usual alphabetic median if the moves (and the pieces removed by the opponent) are heavily biased to a particular subset of player-pieces. It is conceivable that an opponent could identify the AlphaMove strategy as being used, and then use their foreknowledge of the algorithm's 'developing game' to strategically make (normally non-optimal) moves designed explicitly to force the algorithm down their own choice of path, such as targeting undefended rooks and queens (either capturing them with impunity, or just strategically restricting their movements by moving into contact with them in such a way as to normally be a suicidal sacrifice), in order to make certain other pieces take their own moves. Although given the established failings of uncritically sticking to the algorithmic plan, it is probably ''vastly'' more effort to precisely engineer a given game-state than to merely play properly and respond with half-decent responses to the overwhelmingly sub-optimal series of moves. For example, setting Stockfish (Black) against AlphaMove (White) results in the following fifteen-move victory for black:
# f3 e5
# e4 Bc5
# d4 Bxd4
# f4 d5
# g4 Qh4+
# Ke2 Qf2+
# Kd3 dxe4+
# Kxe4 Nf6+
# Kd3 e4+
# Kc4 Be6+
# Kb5 a6+
# Ka5 Bc3+
# Ka4 b5+
# Ka3 Qc5+
# b4 Qxb4#
On the other hand, a person playing black who knew that white was using AlphaMove could win in six moves (possibly fewer):
# f3 e5
# e4 Qh4+
# Ke2 b6
# g3 Ba6+
# Ke1 Qh3
# c3 Qxf1#
However, chess tournaments do not award more points for quicker victories, and playing like this would be risky because if White initially uses AlphaMove and Black goes for the six-move checkmate, White could capture Black's undefended queen on move five, revealing the AlphaMove emulation to just be a ruse to get Black to expose a queen. White would then be in a winning position after 5. Bxh3 or 5. Nxh3 and could play smartly for the rest of the game and would likely win, so opting for the fifteen-move mate would likely be safer. Indeed, playing Stockfish against itself after 5. Nxh3 yields the following 41-move win for white:
# f3 e5
# e4 Qh4+
# Ke2 b6
# g3 Ba6+
# Ke1 Qh3
# Nxh3 c5
# f4 Bb7
# Nc3 Nc6
# d3 Be7
# Nb5 Rd8
# Nc7+ Kf8
# Bg2 d6
# Kf2 Nf6
# Rf1 exf4
# Nxf4 Rc8
# Ncd5 Ne5
# Nxf6 Bxf6
# Kg1 h6
# c3 Rd8
# d4 cxd4
# cxd4 Nd7
# Be3 Kg8
# h4 Kh7
# Nh5 Rhg8
# Qf3 Ba6
# Rfc1 Kh8
# Bh3 Bd3
# Bxd7 Bxh4
# Qxf7 Bxe4
# gxh4 Rdf8
# Qe7 Bg6
# Rc7 Rf7
# Nf4 Rxe7
# Nxg6+ Kh7
# Nxe7 Rf8
# Rf1 Rb8
# Rf7 Kh8
# Be6 Kh7
# Bf5+ Kh8
# Ng6+ Kh7
# Rxg7#
(While white technically does let black capture white's undefended queen on move 34, this is not the same as what black did on move 5 because after 34. … Rxe7, the rook doing the capturing has moved into a position where 35. Nxg6+ forks it and black's king while capturing a bishop. This allows white to recoup {{w|Chess_piece_relative_value|eight points}} in the following two moves, which represents a greater proportion of black's remaining firepower than the nine points that white just lost. Indeed, with black having only one piece (besides the king) left to help with defense, white can force checkmate in six moves. Thus, the AlphaMove ruse, where white pretends to use AlphaMove in order to trick black into hanging the black queen, can be advantageous for white if black falls for it. However, even after 1. f3 e5 2. e4, white has a weakened kingside early in the game and has not been the most productive at allowing for future development, so black should focus on development (e.g., 2. … Nc6 or 2. … Bc5) and use the advantage that black already has instead of risking falling into a trap. (Of course, the AlphaMove ruse would only have any chance of working if the opponent reads xkcd, which is another reason to not try it.)

Even if AlphaMove ever found one or more of its potential moves to be one that happens to result in checkmate, it has no reason to do anything other than choose its "mid-list move", as described, and the chances are high that such a mate would never be invoked. Along those lines, Qa4+ is a relatively safe move to create a short-term check, to put immediate pressure upon the Black king, and potentially a longer term inconvenience with 'only' a predictable response preventing it from developing into a mate. But it is not in the right list position to attempt, never mind whether it would then be correctly followed up.

This engine may be named for and inspired by the real chess engine {{w|AlphaZero}}, or {{w|AlphaGo}} which plays a different game but has a more similar name. Another real name, mentioned in the title text, is {{w|Stockfish (chess)|Stockfish}}, a widely used (and powerful) chess engine.

On this board, Black can win the game instantly with ...Bb4{{w|Checkmate|#}}. Rather than do anything to defend against this, White just moves an unrelated piece, almost certainly losing immediately afterward. Randall has also chosen a setup where the king is placed in a position where it cannot make any legal moves, thus removing it from the list of pieces that can perform any moves. Almost certainly this was a choice, both to make the list without king moves and also to make it pretty easy to see how one more move would be checkmate.

The title of the comic is a play on words. As the name of the chess engine, it refers to the strategy of choosing moves based on alphabetical ordering, while in popular usage, an "alpha move" is an action that would assert dominance over someone else. This makes it an ironic name for the chess engine; rather than asserting dominance, it loses quickly.

The title text mentions games with "six knights", which implies that two pawns have been promoted to knights. Pawns can promote to bishop, knight, queen or rook, so the middle of this list is tied between knight and queen. It is rare that a pawn is promoted to a knight; in most situations a queen would be preferred. The exceptions (perhaps where promoting to queen would cause either an immediate stalemate from what was a winnable position, or let the king survive on a square that's a knight's move away from the newly-promoted queen) are common in contrived chess puzzles but rare in actual gameplay. Promotion to rook or bishop would be even rarer, as these pieces have fewer move possibilities than a queen without the alternative moves of the knight. It would probably be due to a 'forced' promotion, in lieu of moving a more vital piece currently perfectly positioned for the endgame, and the player concerned should be perfectly aware of they must or must not promote to, ahead of time. However, when AlphaMove plays itself, the pawns tend to preferentially move up (and down) the right-hand side of the board, and may meet each other in double-files, or more, ''possibly'' with the opportunity to cross over to capture an opponent. If the alphabetical balance of moves becomes such that multiple right-hand files of pawns are dodging past each other, it is not unrealistic that multiple pawns would reach the respective back ranks and 'choose' to promote to knights.

Chess is a [[:Category:Chess|recurring theme]] on xkcd, with another recent example being [[3036: Chess Zoo]].

==Transcript==
:[A standard chessboard is shown with Black at the top. The boards "black squares" are light gray. Black (which is drawn as dark gray) has made moves resulting in Qf2, Nd4, e4, a5, Bc5, e5 and Ne7 while other black pieces are in starting positions. White has made moves resulting in c4, f4, h4, Kc3, Ne2 and three white pawns are removed from the board while other white pieces including a- and b-pawns are in starting positions. Two squares associated with white's move Ne2 are highlighted in yellow, it has moved there from its starting position g1.]

:[To the right of the chess board is a vertical list of possible moves listed in alphabetical order. The text is in light gray, except the move Ne2 in the middle which is black and highlighted in yellow. Two light gray double arrows with a line at the end of each arrow head goes from the top to just above the yellow move, and from just below this to the bottom. A short but thick black arrow points in between the space between the two gray arrows pointing at the yellow move.]
:a3
:a4
:b3
:b4
:Bd2
:Bd3
:Be2
:Be3
:Bg2
:Bh3
:f5
:fxe5
:h5
:Na3
:Nd2
: '''Ne2'''
:Nf3
:Nh3
:Qa4+
:Qb3
:Qc2
:Qd2
:Qd3
:Qe1
:Qe2
:Qf3
:Qg4
:Qh5
:Qxd4
:Rh2
:Rh3

:[Caption below the panel:]
:My new AlphaMove chess engine, which sorts the list of legal moves alphabetically and picks the middle one, was quickly defeated by stronger engines.

{{comic discussion}}

[[Category:Comics with color]]
[[Category:Comics with lowercase text]]
[[Category:Chess]]

Talk:2997: Solar Protons

2024-10-11T18:22:52Z

172.70.210.235:

cute and wholesome 😊 [[User:CalibansCreations|'''Caliban''']] ([[User talk:CalibansCreations|talk]]) 17:29, 11 October 2024 (UTC)

Re, "idealized depiction": does this mean it's not a real photo? Is it AI? It would be nice to know the source, if possible. [[Special:Contributions/162.158.175.99|162.158.175.99]] 17:57, 11 October 2024 (UTC)

3D stick figure is peak blursed [[Special:Contributions/172.70.210.235|172.70.210.235]] 18:22, 11 October 2024 (UTC)

Talk:2957: A Crossword Puzzle

2024-07-11T00:54:52Z

172.70.210.235:

compare https://www.buttersafe.com/2011/02/17/crosswords/ --[[Special:Contributions/162.158.158.236|162.158.158.236]] 20:50, 10 July 2024 (UTC)

it's *A* crossword puzzle for a reason ;) -- 21:05, 10 July 2024 (UTC)

i’m trying to table-ify it but i keep getting edit conflicted. [[Special:Contributions/172.71.30.93|172.71.30.93]] 21:24, 10 July 2024 (UTC)

Surprised something like "Jagged and loose Hawaiian lava flow (2)" couldn't be fit in (unless I've missed it). Maybe because there were no two-letter answers at all, of course. (I think... Again, maybe I'm missing them.) [[Special:Contributions/172.70.86.38|172.70.86.38]] 21:30, 10 July 2024 (UTC)

unfortunate that [https://tmbw.net/wiki/Aaa "antepenultimate track of They Might Be Giants' ''Glean''"] did not make it in --[[Special:Contributions/172.70.230.200|172.70.230.200]] 21:35, 10 July 2024 (UTC)
:And where is "Fonzie's catch-phrase"? Or does that end with a Y? [[User:Barmar|Barmar]] ([[User talk:Barmar|talk]]) 23:02, 10 July 2024 (UTC)

Did I use the calculator wrong, or 12356631 in base 26 equals 111111, not AAAAAA? [[Special:Contributions/172.69.90.180|172.69.90.180]] 22:33, 10 July 2024 (UTC)
:anyone using base 26 is probably likely to be using all 26 letters, instead of ten numbers and sixteen letters. contextless, i would usually assume any base has standard decimal digits, but liberties have already been taken here so why not (please sign)
::I wasn’t sure enough to comment, but it looks like he miscalculated. 26^5 + 26^4 + 26^3 + 26^2 + 26^1 + 26^0 = 12355631 = 111111 in base 26. To be AAAAAA it would have to be 123556310. Of course, maybe he’s using A through Z instead of the expected 0 through 9 followed by letters A through P, the way hexadecimal is. [[Special:Contributions/172.70.210.52|172.70.210.52]] 23:16, 10 July 2024 (UTC)
:::If he’s using the letters A through Z as the ‘digits’ for base 26, then he’s still wrong, because A would be 0, Z would be 25, and 12355631 decimal would be BBBBBB in that base 26. [[Special:Contributions/172.70.210.235|172.70.210.235]] 00:54, 11 July 2024 (UTC)

I'm surprised he didn't make this interactive, so you could type into all the cells to fill out the crossword. [[User:Barmar|Barmar]] ([[User talk:Barmar|talk]]) 23:02, 10 July 2024 (UTC)

2936: Exponential Growth

2024-05-23T01:29:53Z

172.70.210.235: /* Transcript */ more cats

{{comic
| number = 2936
| date = May 22, 2024
| title = Exponential Growth
| image = exponential_growth_2x.png
| imagesize = 545x264px
| noexpand = true
| titletext = Karpov's construction of a series of increasingly large rice cookers led to a protracted deadlock, but exponential growth won in the end.
}}

==Explanation==
{{incomplete|Created by a 2^64TH ITERATION OF A BOT - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

{{w|Exponential growth}} is the principle that if you keep multiplying a number by a value larger than 1, you will pretty quickly get very large numbers. Even if you start with 1 and simply double it each time, you'll have a 10-digit number after about 30 iterations.

This principle is often illustrated using the story "Game of Rice". A king of India wished to reward a man for creating a new game of Chess, and told him that he'd grant any wish. The man simply asked for a {{w|Wheat and chessboard problem|grain of wheat to be placed on a chess board and for it to double with each square on the board each day.}} The king granted his strange request and ordered one wheat grain to be placed on the board. The second day two more pieces were placed on the square next to that and the day after four pieces on the next. However, by day 20 there was over 500,000 grains on the board. The king had to dig into his own stock pile to pay his dues. On day 24 the king owed 8 million grains. By day 32 the king owed over 2 billion pieces of grain, at this point he had to give up and offered the man another prize.

Black Hat begins describing the metaphor, only to reveal it wasn't a metaphor at all. Black Hat had been playing actual Chess games, and tried to force his opponent to resign by burying the chess pieces in rice.

{{w|Garry Kasparov}} is a world renowned Russian chess master. He had the highest FIDE chess rating in the world-one of 2851 points-until {{w|Magnus Carlsen}} surpassed that in 2013 by 31 points. The Kasparov gambit is an opening move in chess.

In 1984-85 Garry Kasparov played {{w|Anatoly Karpov}} in a 5-month-long 48-game championship tournament which was abandoned. In the 1986 rematch Garry Kasparov retained his world championship title.

In the 1984-85 match Kasparov was losing 4-0 with 6 points being required to win. Kasparov proceeded to draw 35 times before the match was abandoned.

Instead of this being a (possibly apocryphal) story, [[Black Hat]] used it literally during a game of chess to annoy his opponent into quitting.

* First row:
** a1: 1
** a2: 2
** a3: 4
** a4: 8
** a5: 16
** a6: 32
** a7: 64
** a8: 128
* Second row
** b1: 256
** b2: 512
** b3: 1,024
** b4: 2,048
** b5: 4,096
** b6: 8,192
** b7: 16,384
** b8: 32,768
:
* First of each row
:
** c1: 65,536
** d1: 16,777,216
** e1: 4,294,967,300
** f1: 1,099,511,630,000
** g1: 281,474,977,000,000
:
* ...
:
* Eighth row
** h1: 72,057,594,040,000,000
** h2: 144,115,188,100,000,000
** h3: 288,230,376,200,000,000
** h4: 576,460,752,300,000,000
** h5: 1,152,921,505,000,000,000
** h6: 2,305,843,009,000,000,000
** h7: 4,611,686,018,000,000,000
** h8: 9,223,372,037,000,000,000

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[Black Hat is talking to Cueball standing next to him.]
:Black Hat: Exponential growth is very powerful.

:[Closeup on Black Hat. Next to him is an image of the lower left part of a chessboard. The four leftmost squares in the bottom row have grains of rice on them -- one, two, four, and eight grains respectively.]
:Black Hat: A chessboard has 64 squares.
:Black Hat: Say you put one grain of rice on the first square, then two grains on the second, then four, then eight, doubling each time.

:[Black Hat has emptied a bag of rice on a chessboard. There are several bags next to him and a pile of rice already on the table. A frustrated Hairy is walking away, fists clenched.]
:[Caption above panel, representing Black Hat continuing to speak:]
:If you keep this up, your opponent will resign in frustration.
:It's called Kasparov's Grain Gambit. Nearly impossible to counter.

{{comic discussion}}

[[Category:Comics featuring Black Hat]]
[[Category:Comics featuring Cueball]]
[[Category:Comics featuring Hairy]]
[[Category:Chess]]
[[Category:Food]]
[[Category:Comics featuring real people]]

2936: Exponential Growth

2024-05-23T01:26:39Z

172.70.210.235: /* Explanation */ wheat?

Talk:2902: Ice Core

2024-03-04T23:11:53Z

172.70.210.235: Added comment

Akin to [[2729: Planet Killer Comet Margarita]], which perhaps needs mentioning in the upcoming Explanation... [[Special:Contributions/162.158.74.118|162.158.74.118]] 23:04, 4 March 2024 (UTC)

Added a short explanation, but it'll definitely need more work. [[Special:Contributions/172.70.210.235|172.70.210.235]] 23:11, 4 March 2024 (UTC)