The Scottish BookMathematicians work in a rather specific way and - quite often - they require specific conditions to work efficiently. Among these conditions we should mention an appropriate, to a considerable extent public place with a sufficient supply of all kinds of beverage. In other words, something we might call a caf\'e. The Lvov School of Mathematics of the pre-war period was formed by two such neighbouring places: the Scottish Caf\'e (Caf\'e Szkocka) and Caf\'e Roma. Nevertheless, the posterity will only be aware of the first of them due to a lucky investment of its owner. The investment took the form of a thick, carefully bound (and equally carefully attached to the table) notebook where gentlemen mathematicians could lay down their precious ideas, thus misusing fewer paper napkins. Gentlemen mathematicians liked the idea and thus a unique piece of mathematical literature saw the light of day - a collection of more than 193 random mathematical problems being a result of what might be called social life. It should be noticed that both the problems and the answers or comments were written down in various languages (like English or Russian) which happened to occur to the authors at the moment of writing. Sometimes the formulation of a problem was followed by the promise of a prize for the solution. This could be 5 beers or a live goose. The prize was always delivered to the winner.
Stefan Banach was the first to enter a problem to the Book on July 17 of 1935. The last problem, problem 193, was due to Hugo Steinhaus and it bears the date of May 31, 1941. The total number of problems was actually greater than 193, since the numeration used to be repetitive. For instance, there was problem number 10.1, 15.1 or 17.1. Most problems have been solved, though not all of them. In some cases the solution was not a mere intellectual exercise or sport, for it marked the beginning of a new direction of research.
I mentioned that the Scottish Book was written in different languages. It should be clear, however, that the prevailing language was Polish. And here is an example of what we can do with our own tradition: the Scottish Book was published by... the Boston branch of the Birkh\"auser publishing house. The publication had been edited by R. Daniel Moulding and Polish mathematicians had a great share in the edition. The Book contains - besides some comments on the previous editions of its problems - five lectures delivered at a conference on the Book (by Stanis\l aw Ulam, Marek Kac, Antoni Zygmund, Paul Erd\"os and Andrzej Granas) and all the original problems with very interesting comments by more than fifty mathematicians, mostly Polish. I am giving a description of the Book here, for I am afraid that for most of my readers there is little hope for ever seeing it. That's a pity; they would have had the chance of seeing the facsimile of several pages of the original Scottish Book. Anyway, let's stop whetting the appetite.
I shall now quote six problems drawn from the Scottish Book. Three of them have been solved, the other three have not.
Problem 152 (Steinhaus)
A circle with radius 1 contains at least two points with integer coordinates (x,y) and at most five such points. If one translates this circle by vectors nw (n=1,2,3,...), where w=(a,b) has both coordinates irrational and such that their ratio a/b is also an irrational number, then numbers 2, 3, 4 [ of points with both coordinates being integers ] will appear infinitely many times. What is the frequency of their appearing as n goes to infinity? Does it exist?
Problem 44 (Steinhaus)
A continuous function z=f(x,y) describes a surface with the property that two straight lines lying entirely on the surface pass through each of its points. Prove that the surface is a hyperbolic paraboloid. Prove the same without the assumption of continuity.
In the Scottish Book the problem is followed by a
Note: The problem has been solved in the positive by Stefan Banach - also without the assumption of continuity. The proof is based on the following observation: any two straight lines lying on such a surface either intersect or their projections onto the plane xy are parallel.
July 30, 1935.
I invite you to follow Banach's way.
Problem 59 (Ruziewicz)
Can a square be decomposed into a finite number of squares of pairwise different size?
Problem 60 (Ruziewicz)
Given a positive number epsilon, can the surface of a unit sphere be divided into finitely many connected and pairwise congruent components so that the diameter of each is less than epsilon?
Again, both the Mathematical Kaleidoscope and Delta mention another problem:
Problem 19 (Ulam)
Must a solid of homogeneous density which can float on water in arbitrary position be a sphere?
Here is a similar problem formulated just few years ago:
At what amount of liquid in a bottle is the center of gravity at its lowest position?
The problem has practical significance, namely, when to make a break in drinking during a journey. I shall present no solution here, even if I know one. Let the readers enjoy the puzzle too.
This example may suggest that the problems in the Scottish Book are shear entertainment. This is not so, though. It's just me choosing the simplest. Here is another. This time it uses a very scientific language, which comes as no surprise if we consider the long list of authors.
Problem 10.1 (Mazur, Auerbach, Ulam, Banach)
Theorem. If K(n) (n=1,2,3,...) is a sequence of convex bodies such that each of them has diameter smaller than or equal to a and the sum of their volumes is smaller than or equal to b, then there exists a cube of diameter c=f(a,b) with the property that all the bodies can be disjointedly collocated within it.
Corollary. A kilogram of potatoes can be collocated within a bag of finite dimensions.
Problem. Determine the function c=f(a,b).