Scottish Book Problems

The Scottish Book problems are the mathematical questions and recorded topics collected by mathematicians meeting at the Scottish Café in Lwów, now Lviv, Ukraine. The original was a bound handwritten notebook. It began in 1935, and its last dated contribution is from 1941. Its problems were contributed by several mathematicians, including Banach, Mazur, Steinhaus, and Ulam (Ulam 1958, Mauldin 2015).

The photograph shows facsimile pages displayed at an exhibition, not the original bound notebook. The photograph by Stako is licensed under Creative Commons Attribution-ShareAlike 3.0.

The collection includes problems in functional analysis, topology, set theory, and geometry. Problem 19, the Ulam floating body problem, asks whether a homogeneous convex body floating in equilibrium in every orientation must be a closed ball. Ryabogin (2022) answered this negatively using bodies of relative density . The original problem statements are distinct from the later solutions and commentaries included in Mauldin's (2015) expanded edition, which also contains selected problems from the New Scottish Book.

The Scottish Book should also be distinguished from Ulam's (1960) A Collection of Mathematical Problems, which includes material originating both in the Scottish Book and in Ulam's later work.

The New Scottish Book continued the tradition in postwar Wrocław at the initiative of Edward Marczewski. Hugo Steinhaus entered its first problems in July 1946. Its three volumes contain 968 problems (University of Wrocław). The photograph is from exhibition press materials of the Memory and Future Centre, published by Culture.pl.

The following table summarizes all 193 numbered problems of the original Scottish Book in their original order, together with the five decimal-numbered additions 10.1, 15.1, 17.1, 20.1, and 188.1 (Mauldin 2015). The descriptions summarize the original statements rather than the later solutions or commentary. Problem 171 is misprinted as a second Problem 163 in the 2015 edition. The proposer of Problem 189 is listed as Fermant, following that edition.

number	proposer	problem
1	Banach	compact remetrization of metric spaces preserving convergent sequences
2	Banach and Ulam	congruence-invariant finitely additive measures on compact metric spaces and uniqueness of equal-piece decompositions
3	Banach and Ulam	impossibility of a compact set being congruent to a proper subset of itself
4	Schreier	Cesàro summability of almost every subsequence of a bounded Cesàro-summable sequence
5	Mazur	comparison of density-one subsequence convergence with matrix summability methods, including bounded sequences
6	Mazur and Orlicz	normal forms for invertible row-finite infinite matrices in summability theory
7	Mazur and Banach	homeomorphism types of infinite-dimensional convex subsets of Banach spaces
8	Mazur	representing Cesàro-summable series as Cauchy products of convergent series
9	Mazur and Orlicz	common intersections of families of finite sets of uniformly bounded size
10	Banach and Mazur	representation of continuous linear functionals on spaces with pointwise convergence by finitely many evaluations
10.1	Mazur, Auerbach, Ulam, and Banach	bounded-cube packing bounds for countably many convex bodies with bounded diameters and finite total volume
11	Banach and Ulam	product extensions of finitely additive measures on integers and measurability of convergence, coprimality, and order relations
12	Banach	ambient equivalence of embedded spheres to standard spheres under tangent-plane hypotheses
13	Ulam	lifting Boolean operations on subsets of integers modulo finite sets to maps of integers
14	Schauder and Mazur	polynomial approximation of functions and their weak derivatives in
15	Schauder	integrability thresholds for solutions of the Poisson's equation with zero boundary values in arbitrary dimension
15.1	Mazur and Orlicz	whether local boundedness forces a complete metrizable topological vector space to be normable
16	Ulam	measures on spaces of measurable functions with prescribed product laws for point evaluations
17	Ulam	a converse law of large numbers for Bernoulli trials with varying success probabilities
17.1	Ulam	agreement of every continuous function with an analytic function on some perfect set
18	Ulam	existence of knotted magnetic field lines generated by current in a knotted closed wire
19	Ulam	Ulam floating body problem
20	Ulam	approximation of homeomorphisms by compositions of coordinate-preserving homeomorphisms
20.1	Mazur and Orlicz	minimal and uniformly bounded numbers of points for irreducibility-preserving linear substitutions
21	Ulam	maps from a disk onto a torus with arbitrarily small fibers
22	Ulam and Schreier	generating all subsets of the real line from additive subgroups by countable unions and set differences
23	Schauder	interior regularity and higher-dimensional extensions of boundary estimates for saddle functions
24	Mazur	continuity of additive functionals whose compositions with continuous paths are measurable
25	Schauder	operator classes extending singular integral equations beyond the classical Fredholm setting
26	Mazur and Orlicz	whether uniform convergence of linear functionals on bounded sets implies uniform convergence near zero
27	Mazur and Orlicz	common zeros of complex polynomials on Banach spaces from bounded approximate common zeros
28	Mazur	possible Cesàro sums of rearrangements of a real series and analogs for other summation methods
29	Ulam	simplicity of identity components of homeomorphism groups of spheres
30	Ulam	simultaneous conjugacy of pairs from conjugacy of corresponding group words
31	Ulam	whether conjugacy classes in compact metric groups are of first category
32	Ulam	approximation of multiplication and inversion in compact metric groups by finite group laws
33	Ulam	equivalence of sequences of projective or measurable sets to sequences of Borel sets under bijections
34	Ulam	whether a collection closed under countable unions and set differences and selections must contain every subset of the real line
35	Ulam	homeomorphism of infinite-dimensional projective Hilbert space with Hilbert space
36	Ulam	retraction of the Hilbert-space unit ball onto its boundary sphere
37	Ulam	classification of rings of sets and sigma-rings of sets up to isomorphism
38	Ulam	limiting connectivity probabilities in a random friendship model with a fixed number of choices per person
39	Auerbach	discontinuous multiplicative, subadditive analogs of absolute value
40	Banach and Ulam	extension of Lebesgue measure to all projective sets while preserving congruence invariance
41	Mazur	finite-dimensional Banach spaces universal for all Banach spaces of a fixed dimension
42	Ulam	fixed points for continuous transformations of families of compact convex sets
43	Mazur	Banach-Mazur game and variants defined by binary digits or decreasing positive terms
44	Steinhaus	classification of surfaces carrying two straight lines through every point, with and without continuity
45	Banach	automatic continuity of Baire-measurable products of homomorphisms of complete metric non-Abelian groups
46	Banach	unicoherence of spheres in Banach spaces
47	Banach	decomposition of permutations of an infinite matrix into row-preserving and column-preserving permutations
48	Mazur and Banach	isomorphism between spaces of continuous functions on countable compact sets and spaces of convergent sequences
49	Mazur and Banach	universal Banach spaces in the separable reflexive, countable-basis, and separable-dual classes
50	Banach	Baire measurability of the Denjoy integral as a functional
51	Mazur	countability of pairwise orthogonal families of measurable functions or numerical sequences without square-integrability assumptions
52	Banach	descriptive complexity of the set of everywhere-differentiable continuous functions
53	Banach	whether uniform short-arc connectivity of an embedded disk forces finite area and almost-everywhere tangent planes
54	Schauder	extensions of the Schauder fixed point theorem to more general topological vector spaces
55	Mazur	factorization of polynomials bounded near unbounded sets through affine functionals bounded on those sets
56	Mazur and Orlicz	simultaneous pointwise blow-up under vanishing translations for discontinuous polynomial-type functionals
57	Ruziewicz	continuous functions with prescribed oscillation bounds and arbitrarily large normalized difference quotients
58	Ruziewicz	finite and infinite chains of immediate successors among topological embeddability types
59	Ruziewicz	perfect square dissection into pairwise unequal squares
60	Ruziewicz	partitioning a sphere into arbitrarily small congruent connected regions with controlled boundaries
61	Steinhaus	surfaces containing intersecting congruent plane curves through each point, with local or global congruence requirements
62	Mazur and Ulam	whether an increasing union of mutually isomorphic groups is isomorphic to its constituent groups
63	Mazur and Ulam	inheritance of irredundant generating sets by subgroups
64	Mazur	separation of positively separated convex bodies in Banach spaces by hyperplanes
65	Mazur	preservation of nowhere density when a convex set is symmetrized
66	Mazur	almost-everywhere existence of mixed derivatives from pure second derivatives and higher-dimensional analogs
67	Banach	winning strategies in cardinality-halving variants of Banach-Mazur game
68	Ulam	convexity of manifolds whose hyperplane sections of the boundary are spheres
69	Mazur and Ulam	metric characterization of Banach spaces by unique iterated metric centers and their extensions
70	Ulam	polynomial approximation with quantitative derivative separation, motivated by Hilbert's fifth problem
71	Ulam	permutations of the natural numbers preserving asymptotic density of every subset
72	Mazur	characterizing complete metrizable vector spaces whose compact sets have compact closed convex hulls
73	Mazur and Orlicz	sharp polarization constants comparing symmetric multilinear operators with their diagonal polynomials
74	Mazur and Orlicz	coefficient bounds for homogeneous polynomials bounded on the unit ball
75	Mazur	preservation of polynomial boundedness near an unbounded set under scalar dilation
76	Mazur	convexity of the body whose radial function is the area of central sections of a convex body
77	Ulam	cancellation of Cartesian squares for homeomorphism, isometry, and group isomorphism
78	Steinhaus	classification of surfaces carrying two prescribed families of congruent curves through each point
79	Mazur and Orlicz	whether polynomial bijections with polynomial inverses must be affine
80	Mazur	changes of Borel class under countable products and eventually constant sequence constructions
81	Steinhaus	characterization of surfaces generated in two ways by congruent curves and possible additive separation of their equations
82	Steinhaus	countability, orthogonality, and completeness of families of mutually independent measurable functions
83	Auerbach	whether pointwise fractional-order difference bounds imply a global Hölder condition
84	Auerbach	whether projective equivalence of all central plane sections characterizes ellipsoids
85	Banach	complete orthonormal systems for which expansions of polynomials or analytic functions diverge almost everywhere
86	Banach	completion of uniformly bounded orthonormal systems without increasing their common bound
87	Banach	intermediate bounds for nonlinear Lipschitz operators
88	Mazur	weighted absolute summability of coefficients when the associated Hankel matrix maps bounded sequences into
89	Mazur	whether strict convexity forces weak convergence to a boundary point to be norm convergence
90	Ulam and Auerbach	the minimum number of elements generating a dense subgroup of a semisimple Lie group
91	Mazur	whether a finite-dimensional normed space isometric to its dual must be Euclidean
92	Mazur	weighted summation of bounded sequences with nonincreasing or completely monotone weights, and uniqueness of resulting limits
93	Mazur	continuous bounded-variation parametrizations of curves and attainment of their minimum possible length
94	Lomnicki and Ulam	an integral limit related to a converse law of large numbers
95	Schreier and Ulam	embedding the additive group of real numbers in the permutation group of a countable set
96	Ulam	compact or locally compact metrizable group topologies on the full permutation group of the integers
97	Kuratowski and Ulam	whether mutual approximation by maps with arbitrarily small fibers forces manifolds to be homeomorphic
98	Schreier and Ulam	finite smooth generating families approximating all continuous maps or homeomorphisms of a sphere
99	Ulam	generation of subsets of the square from Cartesian product sets by countable unions and set differences
100	Ulam and Banach	closed subsets of spheres obtainable as images of limits of homeomorphisms
101	Ulam	whether transitivity on infinite coinfinite subsets forces a permutation group to be the full symmetric group
102	Ulam	comparison of encounter probabilities for one or two moving points, and analogous first-approach questions for circle rotations
103	Schreier and Ulam	a separable topological group universal for locally compact groups
104	Schauder	existence of minimizers for regular nonparametric variational problems with fixed or free boundary data
105	Schauder	existence and regularity of minimizers for corresponding parametric surface variational problems
106	Banach	affine structure of the set of sums of convergent rearrangements of a Banach-space series
107	Sternbach	fixed points of continuous self-maps and homeomorphisms of nonseparating plane continua
108	Banach, Mazur, and Ulam	choosing Schauder bases from prescribed dense subsets or dense linear subspaces of Banach spaces
109	Mazur and Ulam	whether finitely generated rational or composition-closed classes of functions can be closed under indefinite integration
110	Ulam	fixed points for continuous self-maps with uniformly small iterates on manifolds and continua
111	Schreier	uncountable groups in which every countable subset lies in a finitely generated subgroup
112	Schreier	whether class-preserving group automorphisms must be inner
113	Schreier	classification of composition-preserving automorphisms of the space of continuous real functions
114	Auerbach and Ulam	approximating a sphere by an embedded plane in analogy with a curve spiraling toward a circle
115	Ulam	homeomorphisms of Euclidean spaces with dense orbits and possible density of nearly every orbit
116	Schreier and Ulam	density and uniform distribution of random products in compact groups and analogous measure-preserving dynamics
117	Fréchet	differentiable parametrizations with nonvanishing velocity for Jordan curves having oriented tangents everywhere
118	Fréchet	maximal determinants of sign matrices and their asymptotic growth
119	Orlicz	uniformly bounded orthogonal systems giving uniformly convergent expansions of all continuous functions
120	Orlicz	rates of approximation of Hölder-continuous functions by selected monomials
121	Orlicz	everywhere-divergent Fourier series with coefficients in every
122	Mazur and Orlicz	unconditionally convergent but not absolutely convergent series in every infinite-dimensional Banach space
123	Steinhaus	ham sandwich theorem for finite-Lebesgue-measure sets
124	Marcinkiewicz	uniqueness for a convolution integral equation with a continuous kernel nonzero at the origin
125	Infeld	factorization of functions along curves satisfying two first-order differential constraints
126	Kac	vanishing of a characteristic-function limit for mean-zero functions with infinite second moment
127	Kuratowski	representation of closed subsets of zero-dimensional metric spaces as countable intersections of clopen sets
128	Nikliborc	characterization of ellipsoids by polynomial Newtonian potentials inside homogeneous solids
129	Nikliborc	characterization of ellipsoidal shells by constancy of their Newtonian potential in the enclosed cavity
130	Kaczmarz	lower bounds for supremum norms of combinations of lacunary orthogonal functions in terms of coefficient sums
131	Zygmund	existence of Dini integrals under bounded truncated-integral hypotheses
132	Sierpiński	a universal Baire function of two variables representing every bivariate function by substitutions in one variable
133	Eilenberg	disjoint clopen refinements of clopen covers of metric spaces
134	Eilenberg	whether products of indecomposable continua remain indecomposable
135	Eilenberg	invariance of non-unicoherence under local homeomorphisms of locally connected continua
136	Eilenberg	dimension-raising open maps
137	Eilenberg	dimension-raising restrictions of globally dimension-lowering maps to closed subsets
138	Eilenberg	absolute-retract properties of compact convex sets in linear and metrically convex spaces
139	Ulam	conjugating injective continuous Euclidean self-maps to maps preserving measure-zero sets
140	Ulam	classification of arbitrary self-maps up to conjugacy by bijections
141	Ulam	measurable conjugacy of irrational rotations of circles and spheres
142	Ulam (theorem: G. Birkhoff)	representation of arbitrary groups as automorphism groups of binary relations
143	Mazur	definability of the inequality predicate from specified arithmetic operations, composition, and recursion
144	Mazur and Ulam	injective maps from separable Banach-space balls into an interval sending open sets to sets of positive measure
145	Ulam	existence of nonatomic countably additive measures on sigma-algebras generated by prescribed sequences of sets
146	Ulam	rates of convergence in the Lebesgue density theorem
147	Auerbach and Mazur	corner-to-corner 45-degree billiard paths in rectangles with rational side ratios
148	Auerbach	connectedness of complements of polynomial zero sets in terms of signs of irreducible factors
149	Nikliborc	comparison of Newtonian surface potentials at a common tangency point of nested convex surfaces
150	Nikliborc	recovery of symmetry of a surface and its density from symmetry of the exterior Newtonian potential
151	Wavre	nonzero harmonic functions vanishing on all cube edges, and a related algebraic-function contour-integral problem
152	Steinhaus	frequencies of lattice-point counts in irrational translates of a unit disk
153	Mazur	uniform approximation of a continuous function by finite sums
154	Mazur	approximation using only nonzero orthogonal-expansion coefficients and universal linear summation methods for continuous functions
155	Mazur and Sternbach	whether bijective local isometries between Banach spaces are global isometries
156	Ward	almost-everywhere regular parametrizations of surfaces possessing geometric tangent planes everywhere
157	Ward	monotonicity from positivity of upper right approximate derivatives of approximately continuous functions
158	Stoïlow	continuous complex functions analytic off perfect discontinuous singularity sets with prescribed image behavior
159	Ruziewicz	universal power-series partial sums approximating families of continuous functions with a uniform truncation bound
160	Mazur	connectedness of complete metric groups generated by arbitrarily small neighborhoods, and arcwise connectedness of connected metric groups
161	Kac	a central limit theorem for lacunary trigonometric sums, especially frequencies
162	Steinhaus	almost-everywhere upper and lower limits of measurable periodic functions evaluated at integer multiples
163	von Neumann	existence of strictly positive countably additive measures on complete Boolean algebras
164	Ulam	uniform bounds on escape times for approximate iterates of maps of finite subsets of an interval
165	Ulam	boundedness of Lipschitz constants in a greedy extension of finite bijections of points with rational coordinates
166	Ulam	distinguishing manifolds by groups of homeomorphisms preserving a continuous function
167	Ulam	simultaneous equality of finitely many continuous function values before and after a self-map of a Hilbert sphere
168	Ulam	countably generated sigma-algebras containing every analytic set
169	Szpilrajn	congruence-invariant finitely additive extensions of Carathéodory length to all plane sets
170	Szpilrajn	absolute measurability of plane sets whose homeomorphic images are all Lebesgue measurable
171	Schreier and Ulam	whether associativity and permutation invariance characterize composition of self-maps
172	Eidelheit	relations between sequential weak-star closure properties and coincidence of weak-star and weak sequential convergence
173	Eidelheit	norm density of invertible bounded linear operators among all bounded operators on a Banach space
174	Eidelheit	Neumann-series expansion of inverses of operators on complete topological vector spaces
175	Borsuk	homeomorphism of products involving the Hilbert cube and a T-shaped curve with the Hilbert cube
176	Eidelheit	approximation of inverses in a unital Banach algebra by polynomials in the original element
177	Kac	functions of two Hermitian matrices that are positive definite for every pair
178	Kac	characterization of two-sided exponential distributions through a harmonic-mean identity for characteristic functions
179	Offord	Littlewood-Offord-type concentration bounds for random signed sums
180	Kampé de Fériet	representation of stationary random functions by a uniform random variable with independence at distinct times
181	Steinhaus	positive continuous or analytic functions whose integer translates sum identically to 1
182	Knaster	effective decompositions of balls into chords and higher-dimensional analogs
183	Bogolubow	linearization of compact connected locally connected groups acting on Euclidean space
184	Saks	nonnegativity of the Laplacian of a subharmonic function with pure second derivatives everywhere
185	Saks	recovery of surface area as a limit of integrals of finite-difference area elements
186	Banach	complete orthonormal systems giving almost-everywhere unbounded expansions of every nonzero continuous function
187	Alexandroff	Alexander duality for images of punctured polyhedra, equivalence of dimension definitions, and dimension-raising open maps
188	Sobolew	sharp differentiability thresholds for local solvability of nonlinear hyperbolic Cauchy problems
188.1	Eidelheit	absolute continuity along absolutely continuous curves for coordinatewise absolutely continuous functions, with or without -integrable partial derivatives
189	Fermant (as printed)	a Bloch-type universal disk bound in the principal star of a normalized analytic function
190	Lusternik	generalized eigenfunctionals corresponding to continuous-spectrum points of self-adjoint operators
191	Szpilrajn	existence of stochastically independent generating families for nonatomic probability measures
192	Knaster and Szpilrajn	comparison of the countable chain condition with Knaster's condition and preservation under products
193	Steinhaus	match-count probabilities and expected and probable numbers for two boxes of fifty matches

The commentary on Problem 153 relates it to the approximation problem. The commentary tentatively identifies Problem 193 with the Banach matchbox problem (Mauldin 2015, pp. 281-282).