Can you put n pennies on an n x n chessboard so that all of the distances are distinct/how to look that up?

 In Jan 2023 I went to the Joint Math Meeting of the AMS and the MAA and took notes on things to look up later. In one of the talks they discussed a problem and indicated that the answer was known, but did not give a reference or a proof. I emailed the authors and got no response. I tried to search the web but could not find it. SO I use this blog post to see if someone either knows the reference or can solve it outright, and either leave the answer in the comments, point to a paper that has the answer in the comments, or email me personally. 

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A chessboard has squares that are 1 by 1. 

Pennies have diameter 1.

QUESTION: 

For which n is there a way to place n pennies on squares of the n x n chessboard so that all of the distances between centers of the pennies are DIFFERENT?

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I have figured out that you CAN do this for n=3,4,5. I THINK the talk said  it cannot be done for n=6. If  you know or find a proof or disproof then please tell me. I am looking for human-readable proofs, not computer proofs.  Similar for higher n.

I have a writeup of the n=3,4,5 cases here (ADDED LATER- I will edit this later in light of the very interesting comments made on this blog entry.) 

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With technology and search engines it SHOULD be easier to find out answers to questions then it was in a prior era. And I think it is. But there are times when you are still better off asking  someone, or in my case blog about it, to find the answer. Here is hoping it works!

ADDED LATER: Within 30 minutes of posting this one of my readers wrote a program and found tha tyou CAN do it for n=6 and gives the answer. Another commenter pointed to a website with the related quetion of putting as many pawns as you can on an 8x8 board.

ADDED LATER: There are now comments on the blog pointing to the FULL SOLUTION to the problem, which one can find here. In summary: 

for n=3,...,7  there IS a way to put n pennies on a chessboard such that all distances are distinct.

for n=8,...,14 a computer search shows that there is no such way.

for n=15 there is an INTERESTING PROOF that there is no such way (good thing - the computer program had not halted yet. I do not know if it every did.) 

for n\ge 16 there is a NICE proof that there IS such way. 

I am ECSTATIC!- I wanted to know the answer and now I do and its easy to understand!

Don't Negotiate with Logic

Computer science and mathematicians often try to use logic to negotiate whether it be at a university or life in general. I've tried it myself and it doesn't usually work. Even if you have that (rare) perfect argument, remember Upton Sinclair's words: It is difficult to get a man to understand something, when his salary depends on his not understanding it.

So make sure their salary depends on them understanding it. Or more to the point, in a world with limited resources, why it makes sense for them to help you. 

1. Ideally go for the win-win. Why a certain decision helps the department/college/university as well as yourself. Asking for a small investment as a seed towards a large grant for example.
2. How would the decision make you or your students more successful? The success of a department is measured by the success of the faculty and students. On the other hand, why would a different decision hold you and your students back.

Even outside the university, make your objectives in line with the objectives of the person you are negotiating with to lead to a better outcome.

Of course sometimes you are haggling over a price or a salary when it really is a zero-sum game. There it's good to know the BATNA, Best Alternative To a Negotiated Agreement, for yourself and the other entity. In other words, if they aren't selling to you what other options do you and they have?

There are whole books written about negotiating strategies. Mostly it comes down to making it work for both parties. That's what matters, not the logic.

Ted Kaczynski was at one time the worlds most famous living mathematician

(This post is inspired by the death of Ted Kaczynski who died on June 10, 2023.) 

From 1978 until 1995 23 mailbombs were sent to various people. 3 caused deaths, the rest caused injuries. The culprit was nicknamed The Unabomber (I wonder if he liked that nickname.) For more on his story see here.

The culprit was Ted Kaczynski. He had a BS in Math from Harvard in Mathematics in 1962, and a PhD in Math from The Univ of Michigan in 1967. He got a job in the Berkeley math dept but resigned in 1969. He soon thereafter moved to a shack in the woods (I wonder if his prison accommodations were better) and began sending out the mailbombs. 

When he was caught in 1995 he was (a) famous and (b) a mathematician. That last point is debatable in that I doubt he was doing math while living in his shack. But we will ignore that point for now. Would you call him a famous mathematician? If so then he was, in 1995, the most famous living mathematician. 

In  1995 Andrew Wiles proved Fermat's Last theorem (this is not quite right- there was a bug and it was fixed with help from Richard Taylor) and he was, for a brief time, the world's most famous living mathematician, though perhaps Wiles and Kaczinski were tied. Wiles made People magazine's 25 most intriguing people of the year! (NOTE- I originally had, incorrectly that Wiles had proven it in 1986. A comment alerted me to the error which makes the story MORE interesting since Ted and Andrew were competing for Most Famous Living Mathemticians!)

Terry Tao won the Fields medal (2006) AND the MacArthur Genius award (2006) AND the breakthrough award (2015). The last one got him a spot on  The Colbert Report (2014) (See here,) For those 15 minutes he might have been the most famous living mathematician. He did not have much competition for the honor.  

And then there is Grigori Perelman who solved the Ponicare Conjecture and declined the Fields Medal and the Millennium prize (Colbert commented on this, see here.) For a very brief time Perelman may have been the most famous living mathematician. He did not have much competition for the honor. 

The most famous mathematicians of all time: Pythagoras of Samos, Euclid, Lewis Carroll. 

1) Pythagoras might not count since its not clear how much he had to do with his theorem.

2) Lewis Carroll is the most interesting case. He IS famous. He DID do Mathematics. He DID mathematics while he wrote the books that made him famous. So he is a famous mathematician but he is not famous for his math. But that does not quite seem right. 

3) The Math version of AND and the English version of AND are different. Lewis Carroll is FAMOUS and Lewis Caroll is A MATHEMATICIAN but it doesn't seem quite right to call him a FAMOUS MATHEMATICIAN. Same for Ted K.  Andrew W was, for a short time, a legit FAMOUS MATHEMATICIAN. 

3) Stephen Hawkings has appeared on ST:TNG and his voice on The Simpsons, Futurama, The Big Bang Theory. He is famous for a combination of his disability, his expository work, and his Physics. Is he a famous actor? 

4) Science expositors like Carl Sagan and Neil deGrasse Tyson are famous for being expositors of science, not quite for their science. How do Professor Proton and Bill Nye the Science Guy fit into this?

5) Looking at Ted K, Andrew W, Terry T, Grigori P one other point comes up: All of them were famous for a short time but it faded QUICKLY. So- fame is fleeting!

Randomized Acceptances

NeurIPS recently released their 2021 consistency report, a sequel to the 2014 experiment. While the conference has grown dramatically, the results remain "consistent", about 23% disagreement from two separated program committee groups. As before I don't find this too surprising--different committee members have different tastes.

Roughly conference submissions fall into three categories

1. Clearly strong papers
2. Clear rejects
3. A bunch that could go either way.

A typical program committee quickly sorts out the first two groups and then painfully spends considerable time arguing over the others.

What if instead we took a different approach. Accept all the strong papers and reject the weak ones. Choose the rest randomly, either with a uniform or weighted distribution based on the ranking. Maybe reduce the probability of those who submit multiple papers.

Choosing randomly reduces biases and can increase diversity, if there is diversity in submissions. Knowing there is randomness in the process allows those with rejected papers to blame the randomness and those whose papers gets in claim they were in the first group. Randomness encourages more submissions and is fair over time.

Note we're just acknowledging the randomness in the process instead of pretending there is a perfect linear order to the papers that only a lengthy program committee discussion can suss out.

We should do the same for grant proposals--all worthy proposals should get a chance to be funded.

I doubt any of this will ever happen. People would rather trust human decisions with all their inconsistencies over pure randomness. 

Posting a book online with a password- that you broadcast.

 Recently Lance, at the request of  Vijay Vazirani, tweeted the following (I paraphrase)

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Online and Matching-based Market Design book now available as free PDF 

cambridge.org/files/9216/848  

The password is OMBMD_CUP

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This tweet raises two questions.

1) When a book is put online, does it cut into sales? I've heard that people still like paper so the posting on line might be like an ad for the book. Also, for academic books, the authors WANT it to be out there and don't care so much about sales.  If Gasarch & Martin's Bounded Queries in Recursion Theory was available for illegal download I would be delighted. And surprised. 

2) SO, they post it to make it more available and generate buzz. So why have a password? Was this a compromise: 

a) We want people to BUY the book so we'll post it with a password and limit access. 

b) We want people to SEE the book since as academics we don't care about sales, and it might generate buzz, so we don't want a password

c) COMPROMISE: Have a password but tell everyone what it is. 

If you can think of a more plausible scenario, leave a comment. 

Quantifiers: To Parenthesize or not to Parenthesize? Matrix of Formula: To Bracket or not to Bracket?

 For the book 

Computational  Intractability: A Guide to Algorithmic Lower Bounds

by Demaine-Gasarch-Hajiaghayi 

(See  here for a link to a first draft.) 

we had to make some choices about which notation to use. One of the least important ones was the following: 

When defining NP, and in a few other places should we use: 

                                                  (\exists y)(\forall y)[B(x,y)]

or 

                                                   \exists x : \forall y : B(x,y)

or 

                                                    something else.

We ended up doing it the second way.  But I wondered, which, if either, is standard. So I looked in many math and theoretical CS books looking for places they used quantifiers. Here is what I found

a) Most papers and books really don't use quantifiers at all!  This surprised me. 

b) When quantifiers are used, they are used in definitions, not theorems. 

c) One exception is in logic when they deal with formulas as objects onto themselves.  For example, the inductive definition of a formula will have a step:

                       If f(x_1,...,x_n) is a formula then (\exists x_i)[f(x_1,...,x_n)] is a formula. 

d) Here is a list of the few places I saw quantifiers used and if they used parenthesis or not. I say if it has parenthesis (abbreviated Parens)  or not, and if the matrix of the formula is in square brackets, no brackets, or something ese. 

Cook's classic paper . Page 154 Parens, no Brackets (1971) 

Stockmeyer's paper where he defines PH.  Page 6 Parens and Brackets  (1976)

Computers and Intractability by Garey & Johnson. Page 164. Parens and Brackets (1979)

Morass-like construction of aleph_2 trees in L by Devlin.  Page 2 Parens and matrix in Parens (1979)

Descriptive Complexity by Immerman. Page 38 Parens no Brackets (1999) 

Bounded Queries in Recursion Theory by Gasarch and Martin. Parens and Brackets  Throughout the book.  (1999)

Complexity Theory from Godel to Feynman by Rudich. No Parens, No Brackets in Def of PH. (2003) 

Parameterized Complexity Theory by Flum & Grohe. Page 81 no Parens and no Brackets. 

Computational Complexity: A Modern Approach by Arora & Barak. Page 40. No Parens No Brackets.(2007) 

Computational Complexity: A Conceptual Prospective by Goldreich.  Page 114 no parents, no brackets (2008) 

On Quantifer Rank Equivalence between linear orders by Siders. On page 417 they use quantifiers to state a theorem, which is unusual. Parens no brackets.

Presburger arithmetic, Rational Generating Functions, and quasi polynomials by Woods. Parens no  Brackets. (2012) 

Who witness's the Witness by Abel et al. On Page 69 (which the pointer takes you to) No Parens, no brackets. Colons between quantifiers (2018).

e) What to make of all this?

First off- the RARITY of the use of quantifiers really surprised me. The only place I saw them used a lot was my book, co-authored with Georgie Martin,  Bounded Queries in Recursion Theory. Perhaps it would have sold better if I didn't use so many quantifiers. Oh well. 

Second off- Later works don't use parens and brackets. This is most clear if you just look at Complexity Theory Books 

Garey & Johnson - 1979- parens and brackets

Flun & Grohe- 1998- no parens and no brackts

Immerman- 1999 - parens but no brackets (this is the one exception) 

Arora & Barack- 2007 no parens and  no brackets

Goldreich-2008- no parens and no brackets

If you have a complexity theory book around that is not on this list, look up the definition of NP and the definition of the Poly Hierarchy and see (a) if they use parens around the quantifiers, and (b) if they use square brackets or no brackets of something else. Please leave a comment about it so I test the conjecture that parenthesis are just so 1979.