Wikipedia:Reference desk/Archives/Mathematics/2023 October 1

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October 1

Proving n-gons congruent

For triangles, we have SSS, SAS, ASA, and AAS. If my knowledge is correct, only SSS cannot be generalized to polygons with any number of sides. That is, for any n-gon, SASAS...SASAS (with n-1 S's,) ASASA...ASASA (with n-1 A's,) and A-A-S-A-S...S-A-S-A-S (with n-1 A's) will prove two n-gons congruent. Is there a way to generalize SSS?? Georgia guy (talk) 01:33, 30 September 2023 (UTC)

I'm not sure about this, but it seems that you need five elements to determine a quadrilateral. Bubba73 ^{You talkin' to me?} 01:39, 30 September 2023 (UTC)

The vertices of an n-gon, embedded in the Cartesian plane, need 2n coordinate values to be fixed. The dimension of the isometry group E(2) is 3, leaving 2n − 3 degrees of freedom. For n = 4, this comes indeed out as 5.  --Lambiam 09:21, 30 September 2023 (UTC)

So for n=12, we need 21 coordinate values to be fixed? Also for n=4, we have 18 possibilities (there are only 4 sides and 4 angles thus there are no SSSSS or AAAAA): SSSSA, SSSAS, SSASS, SSSAA, SSASA, SSAAS, SASSA, SASAS, ASSSA, SSAAA, SASAA, SAASA, SAAAS, ASSAA, ASASA, SAAAA, ASAAA, AASAA, which of these 18 can make these two quadrilaterals congruent? 61.224.153.37 (talk) 07:26, 2 October 2023 (UTC)

The prohibition of pSSqSSr (see below) rules out SSSSA (as well as SSSAS and SSASS, which are the same data rotated). The angles of a quadrilateral sum up to 360°, so if you have 3 As, a 4th A does not add a new constraint. Therefore the last three, SAAAA, ASAAA and AASAA (also the same data rotated) are flabby. Now 12 of the 18 remain: SSSAA, SSASA, SSAAS, SSAAA, SASSA, SASAS, SASAA, SAASA, SAAAS, ASSSA, ASSAA, ASASA. But also of these, several are rotated versions of the same data: SSSAA ≅ SSAAS ≅ ASSSA and SSAAA ≅ SAAAS ≅ ASSAA. Now only 8 cases remain: SSSAA, SSASA, SSAAA, SASSA, SASAS, SASAA, SAASA, ASASA. I suggest you examine them yourself. For some it is not hard to find specific values that allow several non-congruent shapes (for example, for SSASA: 5–4–90°–4–90°). For some others, it is easy to see that they make the shape rigid. I you get stuck on a case, come back.  --Lambiam 18:25, 2 October 2023 (UTC)

If you have pSSqSSr, in which p, q and r can be any sequences of As and Ss, the polygonal chain covered by the segment SqS can be freely reflected through the axis formed by the line connecting the outmost vertices of the segment. So this is verboten; there can be at most one SS or SSS in the whole sequence. Also, pASASAq is rigid iff pASAq is rigid, which may help to simplify the case analysis.  --Lambiam 08:53, 30 September 2023 (UTC)

One way of generalising is it triangulation. This is normally used in computer graphics to turn polygons into triangles which most graphics APIs require. But it can also be used to describe this problem. The number of triangles needed to draw an n-gon ( n ≥ 3) is n − 2. To do so you need n edges, so the sides, and n − 3 internal edges, for a total of 2n − 3. Using "s" for an internal edge SSS generalises like so: SSS, SSSSs, SSSSSss, ... --2A04:4A43:90AF:FAB6:15B7:4107:CBCC:461B (talk) 09:15, 30 September 2023 (UTC)

Does "internal edge" mean diagonal?? Georgia guy (talk) 10:22, 30 September 2023 (UTC)

Yes. When dealing with them in computer graphics you have vertices, edges and triangles. A renderer generally doesn't care which edges are internal, which are boundaries – in fact they can be both if an object is constructed out of smaller polygonal pieces which are join at the edge. So they're all edges.--2A04:4A43:90AF:FAB6:15B7:4107:CBCC:461B (talk) 11:54, 30 September 2023 (UTC)

Does -1 have any square root in split-complex number?

Does -1 have any square root in split-complex number? 36.232.139.28 (talk) 10:56, 30 September 2023 (UTC)

No. $(x+jy)^{2}=x^{2}+y^{2}+2xyj$ with $x,y\in \mathbb {R}$. —Kusma (talk) 11:16, 30 September 2023 (UTC)

OK, does $j$ have any square root in split-complex number? 36.232.139.28 (talk) 11:28, 30 September 2023 (UTC)

Try to find $x+jy$ with $(x+jy)^{2}=j$ and you will see. —Kusma (talk) 11:31, 30 September 2023 (UTC)

This will make x^2+y^2 = 0 but xy = 1/2, thus there are no integer solutions, another question: Split-complex numbers with argument between 45 and 135 degrees do not have square roots? And how many square roots (and more generally, how many n-th roots) does a split-complex number have (in the set of the split-complex numbers)? Besides, in the set of the split-complex numbers, an algebraic equation with degree n has how many solutions? (In the set of the complex numbers, an algebraic equation with degree n always has n solutions, but this is not always true in the set of the split-complex numbers) 61.224.153.37 (talk) 07:32, 2 October 2023 (UTC)

A split complex number $a+jb$ has a split complex square root if and only if $|b|\leq a$. It is not difficult to derive this for yourself. Any nonzero split complex number that has a square root actually has two square roots. —Kusma (talk) 10:15, 3 October 2023 (UTC)

But 1 has four square roots: +-1 and +-j, thus actually split-complex numbers with argument between 45 and 315 degrees do not have square roots. 220.132.230.56 (talk) 15:08, 3 October 2023 (UTC)

Indeed there are no split complex square roots of $a+jb$ if $a<|b|$, which aligns with the arguments you have listed. Of all split complex numbers with square roots, $0$ naturally has $1$ unique square root, all numbers with $b=a>0$ or $b=-a<0$ have precisely $2$ unique square roots ${\frac {\sqrt {2a}}{2}}+{\frac {\sqrt {2a}}{2}}j,-{\frac {\sqrt {2a}}{2}}-{\frac {\sqrt {2a}}{2}}j$ and ${\frac {\sqrt {2a}}{2}}-{\frac {\sqrt {2a}}{2}}j,-{\frac {\sqrt {2a}}{2}}+{\frac {\sqrt {2a}}{2}}j$ respectively, and all other numbers have $4$ unique square roots. GalacticShoe (talk) 15:01, 5 October 2023 (UTC)

So how many n-th roots does a split complex have if n >2? Also, in the set of the split-complex numbers, an algebraic equation with degree n has how many solutions? 220.132.230.56 (talk) 17:26, 5 October 2023 (UTC)

Split-complex numbers have the lovely property that if you write $(x+yj)^{n}=a+bj$, then for $n\geq 0$, $a+b=(x+y)^{n}$ and $a-b=(x-y)^{n}$. On the one hand, this immediately tells you that if you want to calculate powers of $x+yj$, you can just take $a={\frac {(x+y)^{n}+(x-y)^{n}}{2}}$ and $b={\frac {(x+y)^{n}-(x-y)^{n}}{2}}$. On the other hand, inverting the process can give you the roots.

For odd powers it is extraordinarily easy. From $x+y={\sqrt[{n}]{a+b}}$ and $x-y={\sqrt[{n}]{a-b}}$ one immediately gets $x={\frac {{\sqrt[{n}]{a+b}}+{\sqrt[{n}]{a-b}}}{2}}$ and $y={\frac {{\sqrt[{n}]{a+b}}-{\sqrt[{n}]{a-b}}}{2}}$, yielding, for odd positive $n$, the single root ${\sqrt[{n}]{a+bj}}={\frac {{\sqrt[{n}]{a+b}}+{\sqrt[{n}]{a-b}}}{2}}+{\frac {{\sqrt[{n}]{a+b}}-{\sqrt[{n}]{a-b}}}{2}}j$.

For even powers, roots are completely analogous to the $n=2$ case. If $a<|b|$, then since either $a+b$ or $a-b$ is less than $0$, there are no roots. If $a=b=0$ then there is precisely $1$ root. If $b=a>0$ or $b=-a<0$ then there are precisely $2$ unique roots ${\frac {\sqrt[{n}]{2a}}{2}}+{\frac {\sqrt[{n}]{2a}}{2}},-{\frac {\sqrt[{n}]{2a}}{2}}-{\frac {\sqrt[{n}]{2a}}{2}}$ and ${\frac {\sqrt[{n}]{2a}}{2}}-{\frac {\sqrt[{n}]{2a}}{2}},-{\frac {\sqrt[{n}]{2a}}{2}}+{\frac {\sqrt[{n}]{2a}}{2}}$ respectively. For all other cases, i.e. $a>|b|$, there are $4$ unique roots. GalacticShoe (talk) 13:37, 6 October 2023 (UTC)

Domino tiling

It is possible to tile 9/2 dominos to a 3x3 square and there is no straight line from a side to its opposite side, also there is no point with four dominos meet.

However, if we do not use half domino and only use the dominos:

• For which (m,n), it is possible to tile m*n rectangle with mn/2 dominos with no straight line from a side to its opposite side?
• For which (m,n), it is possible to tile m*n rectangle with mn/2 dominos with no point with four dominos meet?
• For which (m,n), it is possible to tile m*n rectangle with mn/2 dominos with no straight line from a side to its opposite side and no point with four dominos meet?

36.232.139.28 (talk) 11:02, 30 September 2023 (UTC)

See second section below this one. 88.111.190.170 (talk) 12:03, 30 September 2023 (UTC)

What are all possible graphs of ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz+j=0?

What are all possible graphs of the quadratic function ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz+j=0 in a 3D space? Also, what are all possible graphs of the cubic function ax^3+bxy^2+cyx^2+dy^3+ex^2+fxy+gy^2+hx+iy+j=0 on a 2D plane? 36.232.139.28 (talk) 11:05, 30 September 2023 (UTC)

For your first question, see quadric. —Kusma (talk) 11:09, 30 September 2023 (UTC)

Put the pieces on a board with no piece threaten each other

If no piece threaten each other:

1. On m*n board, what is the largest k such that k queens can be put? And how many ways can them be put?
2. On m*n board, what is the largest k such that k manns can be put? And how many ways can them be put?
3. On m*n board, what is the largest k such that k rooks can be put? And how many ways can them be put?
4. On m*n board, what is the largest k such that k bishops can be put? And how many ways can them be put?
5. On m*n board, what is the largest k such that k knights can be put? And how many ways can them be put?
6. On m*n board, what is the largest k such that k nightriders can be put? And how many ways can them be put?
7. On m*n board, what is the largest k such that k empresses can be put? And how many ways can them be put?
8. On m*n board, what is the largest k such that k princesses can be put? And how many ways can them be put?
9. On m*n board, what is the largest k such that k amazons can be put? And how many ways can them be put?
10. On m*n board, what is the largest k such that k wazirs can be put? And how many ways can them be put?
11. On m*n board, what is the largest k such that k ferzs can be put? And how many ways can them be put?
12. On m*n board, what is the largest k such that k dabbabas can be put? And how many ways can them be put?
13. On m*n board, what is the largest k such that k alfils can be put? And how many ways can them be put?
14. On m*n board, what is the largest k such that k camels can be put? And how many ways can them be put?
15. On m*n board, what is the largest k such that k zebras can be put? And how many ways can them be put?
16. On m*n board, what is the largest k such that k dragon kings (in Shogi) can be put? And how many ways can them be put?
17. On m*n board, what is the largest k such that k dragon horses (in Shogi) can be put? And how many ways can them be put?
18. On m*n board, what is the largest k such that k champions (in Omega Chess) can be put? And how many ways can them be put?
19. On m*n board, what is the largest k such that k wizards (in Omega Chess) can be put? And how many ways can them be put?
20. On m*n board, what is the largest k such that k wildebeests (in Wildebeest chess) can be put? And how many ways can them be put?
21. On m*n board, what is the largest k such that k kirins (in Chu Shogi) can be put? And how many ways can them be put?
22. On m*n board, what is the largest k such that k phoenixes (in Chu Shogi) can be put? And how many ways can them be put?
23. On m*n board, what is the largest k such that k lions (in Chu Shogi) can be put? And how many ways can them be put?

36.232.139.28 (talk) 11:23, 30 September 2023 (UTC)

2. $\lceil {\frac {m}{2}}\rceil \lceil {\frac {n}{2}}\rceil$, I'm fairly sure, with $1$ placement when $m$ and $n$ are both odd, $2$ placements when one of $m$ and $n$ is odd and the other even, and $4$ placements when $m$ and $n$ are both even.

3. $\min(m,n)$, with $(\max(m,n))_{\min(m,n)}$ placements ($(a)_{b}$ denoting the falling factorial)

7. Same largest $k$ as 3. (put them all in a diagonal line), unsure of number of possible placements.

10. $\lceil {\frac {mn}{2}}\rceil$, with $1$ placement when $m$ and $n$ are both odd, and $2$ placements when at least one of $m$ and $n$ is even.

GalacticShoe (talk) 02:24, 2 October 2023 (UTC)

Also, I remember that there is a fairy chess piece which is a combination of the mann and the knight, but I forget its name, what is the name of it? Also answer the questions above for this fairy chess piece. 36.232.139.28 (talk) 11:25, 30 September 2023 (UTC)

Ouch! Wikipedia:Long-term abuse/Xayahrainie43. 88.111.190.170 (talk) 12:01, 30 September 2023 (UTC)

a fairy chess piece which is a combination of the mann and the knight — Centaur. Mitch Ames (talk) 00:02, 1 October 2023 (UTC)

Schinzel's hypothesis H#An analogue that fails

Why all of the values of x^8+u^3 as x runs over F2[u] are composite? 220.132.230.56 (talk) 09:19, 1 October 2023 (UTC)

The section is in sore need of citations, but presumably it's referring to Swan's paper linked at the end of the article. I skimmed the paper but I didn't see anything relevant to the question at hand, though Corollary 5 seems close. It looks like an interesting paper; Quadratic reciprocity kind of pops out as a corollary to a special case. But I don't know if I have the expertise or patience to follow all the details. Note that if x has an odd number of terms then the expression has an even number of terms, hence it's divisible by 1+u; that leaves the case where x has an even number of terms. The first non-trivial case is to prove that 1+u^3+u^8 is reducible. Note that the statements were added by an anonymous user ([1]), so OR perhaps? --RDBury (talk) 10:58, 1 October 2023 (UTC)

PS. I think I found it; it appears to be the example following Corollary 3, taking m=3. --RDBury (talk) 12:37, 1 October 2023 (UTC)

Does the sum of the reciprocals of all primes starting with 7 and ending with 7 in base 10 diverge?

Does the sum of the reciprocals of all primes starting with 7 and ending with 7 in base 10 diverges? Generally, does the sum of the reciprocals of all primes starting with digit d₁ and ending with digit d₂ in a given base diverges? 220.132.230.56 (talk) 16:49, 1 October 2023 (UTC)

Using the proof mentioned in the analog of this question asked a week back, if you can prove that the sum of the reciprocals of all numbers ending with $d_{2}$ but not starting with $d_{1}$ converges, then there must be infinitely many primes starting with $d_{1}$ and ending with $d_{2}$. GalacticShoe (talk) 17:42, 1 October 2023 (UTC)

I asked this question a week back, but since no one replied this question, thus I asked again. 220.132.230.56 (talk) 17:47, 1 October 2023 (UTC)

I'm pretty sure that the sum of the reciprocals of all numbers ending with $d_{2}$ but not starting with $d_{1}$ would not converge, since the amount of numbers starting with $d_{1}$ is fairly small compared to the amount of numbers not starting with $d_{1}$. The Kempner series, intuitively, can converge because the proportion of numbers not containing the digit $9$ at all tends to $0$, but it is not the case here. Given this, I imagine that one has to take a different approach. GalacticShoe (talk) 01:21, 2 October 2023 (UTC)

So is there still a possibility that (sequence A062334 in the OEIS) is a small set? 220.132.230.56 (talk) 05:11, 2 October 2023 (UTC)

It's entirely possible, yes. GalacticShoe (talk) 15:38, 2 October 2023 (UTC)

But both (sequence A045713 in the OEIS) and (sequence A030432 in the OEIS) are known to be large sets? 220.132.230.56 (talk) 17:34, 3 October 2023 (UTC)

Yes. You can prove the former with a generalization of Bertrand's postulate. The largeness of the latter, and -- as per PMajer's proof below -- the largeness of the set of all primes starting and ending with any digits (assuming valid coprimeness constraints), can be proven using Dirichlet's asymptotic expression for the sum of reciprocals. GalacticShoe (talk) 22:27, 4 October 2023 (UTC)

(edited) Let's denote, for $k\in \mathbb {N}$, $s_{k}$ the sum of reciprocals of all primes $p=7{\text{ mod }}10$ such that $7\cdot 10^{k}\leq p<8\cdot 10^{k}$. You are asking whether $\sum _{k=0}^{\infty }s_{k}$ diverges. The answer is yes. The strong form of Dirichlet's theorem on arithmetic progressions asserts that for co-prime $a,n$ one has $\sum _{p{\text{ prime}},p<x \atop p=a{\text{ mod }}n}{\frac {1}{p}}\sim {\frac {1}{\phi (n)}}\log \log x$. Hence in your case one finds $s_{k}\sim {\frac {1}{4}}\log {\Big (}{\frac {\log 8+k\log 10}{\log 7+k\log 10}}{\Big )}\sim {\frac {\log(8/7)}{4k\log 10}}$, whose sum diverges, by asymptotic comparison with the harmonic series. pma 20:31, 4 October 2023 (UTC)

So you proved that $s_{k}$ diverges? 220.132.230.56 (talk) 13:00, 6 October 2023 (UTC)

Indeed they have, consequently showing that the set of all primes starting and ending with $7$ (more generally, starting and ending with any string of digits, assuming valid coprimeness constraints) is infinite. GalacticShoe (talk) 13:04, 6 October 2023 (UTC)

And thus (sequence A062334 in the OEIS) must be a large set? 220.132.230.56 (talk) 11:11, 7 October 2023 (UTC)

Yup, by definition. GalacticShoe (talk) 01:25, 8 October 2023 (UTC)

Is this true in all bases b? i.e. in any base b, the sum of the reciprocals of the primes starting and ending with any string of digits in base b (assuming valid coprimeness constraints) is infinite? 220.132.230.56 (talk) 19:41, 8 October 2023 (UTC)

Yes, replace $10$ with $b$. GalacticShoe (talk) 00:42, 12 October 2023 (UTC)