How many pieces can a puzzle have?

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Patrick Honner tweeted a few days ago:

My 7 yo pointed out that our 300 half puzzle genuinely comprises 18 x 18 = 324 pieces and I simply don’t know what to own anymore.

— Patrick Honner (@MrHonner) August 7, 2020

Clearly you are going to also with out difficulty procure a 300-half puzzle, as an illustration 15 by 20. And likewise you are going to also procure a 299-half puzzle — that components as 13 occasions 23. However a 301-half puzzle would favor to be 7 by 43, assuming that the pieces produce a ample grid,and that doesn’t seem love a “life like” dimension for a puzzle.

So which numbers could per chance maybe genuinely be life like sizes for puzzles? This is able to per chance maybe even be fine to know if, say, you prefer to know if you occur to’ve all of the brink pieces. (However a 500-half puzzle could per chance maybe even be 20 x 25, or some life like quite increased numbers love 19 x 27 = 513, or 18 x 28 = 504.) It’s a ample puzzle nonetheless.

So let’s say that a bunch is a “puzzle number” if it’s of the produce m times n with m le n le 2m — that is, a puzzle has to have an factor ratio between 1 (square) and 2. (The different of two is arbitrary right here, but any varied fixed could per chance maybe well be more arbitrary.) We can with out difficulty work out the first few puzzle numbers:

2 = 1 times 2, 4 = 2 times 2, 6 = 2 times 3, 8 = 2 times 4, 9 = 3 times 3


which is ample to get them in the OEIS: that’s A071562, outlined as “Numbers n such that the sum of the middle divisors of n (A071090) is not any longer zero.” (What’s a “middle divisor”, you quiz? It’s a divisor of n that’s between sqrt{n/2} and sqrt{2n}.) Titling it that draw appears honest a dinky uncommon to me: I’d have known as it “Numbers such that the series of middle divisors of n (A067742) is nonzero”.

The first 10,000 puzzle numbers have been calculated; the 10,000th is 35,956. There are 43 under 100, 336 under 1,000, and 2,932 under 10,000 — it doesn’t seem that they have contant density. It’s no longer anxious to acquire this extra — there are 26870 under 10^5, 252,756 under 10^6, 2409731 under 10^7, and 23169860 under 10^8. As an instance, in R, you are going to also generate the puzzle numbers as follows. (This obtain a few seconds to speed. Snarl that you just don’t favor to compute any high factorizations.)

N = 10^8


= obtain(0, N)

for (m in 1:ground(sqrt(N))){


max_n = min(2*m,ground(N/m))


a[m*(m:max_n)] = a[m*(m:max_n)]+1


}

puzzle = which(a >= 1)

I had in the beginning belief this sequence had a pure density, because I turned into once handiest trying numbers as a lot as a few hundred in my head, and for the reason that series of middle divisors appears to moderate somewhere round log(2). There’s a ample heuristic for this – a middle divisor of n is somewhere between sqrt{n/2} and sqrt{2n}; the “likelihood” that a bunch is divisible by k is 1/k, so the “expected number” of middle divisors of n is

sum_{k=sqrt{n/2}}^{sqrt{2n}} {1 over k} approx int_{sqrt{n/2}}^{sqrt{2n}} {1 over x} : dx = log sqrt{2n} - log sqrt{n/2} = log sqrt{4} = log 2.

However there ought to be more numbers as you glide additional out which have many middle divisors, and more zeroes. Here is equal to the behavior you look for the scenario of “how some systems can an integer be written as a sum of two squares”. If so the “moderate” is pi/4, but an integer higher than one could per chance maybe even be written as a sum of two squares if and handiest if its high decomposition comprises no high congruent to some mod 4 raised to an weird energy; asymptotically the series of clear integers under x that are the sum of two squares goes love bx/sqrt{log x} for a fixed b (the fixed is the Landau-Ramanujan fixed). That connection could per chance maybe stem from the reality that each sequences are multiplicative. For sums of two squares, that follows from the factorization-primarily based characterization or from the reality that

(a^2 + b^2) (c^2 + d^2) = (ac-bd)^2 + (ad + bc)^2.

For puzzle numbers, it follows from a observation of Franklin T. Adams-Watters in the OEIS entry. So numbers which have a range of things each are inclined to be sums of two squares and to be puzzle numbers in numerous varied systems; as we gather to better numbers these birth “crowding out” the smaller numbers.

The mathematics web has noticed this as a minimum once ahead of. John D. Cook dinner wrote in 2014: Jigsaw puzzles that say they have 1,000 pieces have approximately 1,000 pieces, but per chance no longer exactly 1,000. Jigsaw puzzle pieces are in overall organized in a grid, so the series of pieces along a aspect must be a divisor of the total series of pieces. This means there aren’t very some systems to procure a puzzle with exactly 1,000 pieces, and most have awkward factor ratios.

1000 as 25-by-40 appears life like, but 27 x 37 = 999 or 28 x 36 = 1008 would furthermore work. I purchase that you just wouldn’t genuinely look a 999-half puzzle for the reason that attorneys would claim calling that 1000 turned into once false advertising and marketing. A puzzle weblog indicates that practically all “500-half” puzzles are genuinely 27 x 19 = 513 and most “1000-half” puzzles are 38 x 27 = 1026 – each of these factor ratios are approximations of sqrt{2}. That’s a upright factor ratio to make spend of if you occur to desire the potential to procure each “500-half” and “1000-half” (or more on the total, “N-half” and “2N-half”) versions of the equal puzzle. Equally the A0, A1, … series of paper sizes dilapidated in numerous of the field have that factor ratio and therefore an An half of paper could per chance maybe even be decrease into two A(n+1) pieces.

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