How many disjoint triangular regions can result from drawing n distinct lines in the plane? Kobon triangle problem [wiki]

Specific Question: How many disjoint triangular regions can result from drawing 10 lines in the plane? Best known answer: for 13 lines: 47, for 14 lines <= 54. [ mathworld ] Sequence: 1, 2, 5, 7, 11, 15, 21, ... OPEN [ OEIS ?]

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Is every slice knot a ribbon knot? quanta

Specific Question: Is there a smoothly slice knot which cannot be represented with only ribbon singularities?

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What is the longest length of a simple path with no chords in the n-dimensional hypercube?? xkcd

Specific Question: What is the longest length of a simple path with no chords in the 9-dimensional hypercube? Sequence: 1, 2, 4, 7, 13, 26, 50, 98, ... OPEN [ OEIS ]

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Does a Moore graph with girth 5 and degree 57 exist? Table

Specific Question: Is there a degree 4 and diameter 3 graph with more than the best known example of 41 nodes (and less than the the Moore bound of 53 nodes)? Sequence: 1, 1, 4, 3, 13, 21, ... OPEN [ OEIS ]

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How many distinct line arrangements are there, on the plane, using n distinct lines, up to parametric equivalence? Affine line arrangements [wiki], [ Notices ] (Forcey, page 2) Oriented matroids. See Table 8.1 on page 165: [ Finschi]

Specific Question: How many parametrically distinct line arrangements are there, on the plane, using 8 distinct lines? Best known answer: for 7 lines: 37830 [ youtube ] (N.J.A. Sloane) Sequence: 1, 1, 2, 4, 9, 47, 791, 37830, ... OPEN [ OEIS ]

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How many posets are there, up to isomorphism, with n elements? Finite partially ordered sets [wiki]

Specific Question: How many posets are there with 17 unlabeled elements? Best known answer: for 16 elements: 4483130665195087 Sequence: 1, 1, 2, 5, 16, 63, 318, 2045, ... OPEN [ OEIS ]

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How many plane arrangements are there in R^3 using n planes? Affine hyperplane arrangements [wiki]

Specific Question: How many plane arrangements are there in R^3 using 8 planes? [stack exchange] 14 for four planes [illustration] Best known answer: 1472049 for 7 planes Sequence: 1, 1, 2, 5, 14, 74, 1854, 1472049 ... OPEN

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What is the smallest number of people such that either k of them are mutual friends or k of them are mutual strangers? Ramsey theory [wiki] What is the smallest complete graph K_n with edges colored (red or blue) such that there is either a red or a blue complete subgraph of size k? We know that n = R(k,k) exists for any k.

Specific Question: What is R(5,5)? [wiki] Best known answers: R(4,4) = 18, and 43 <= R(5,5) <= 48. Conjectured Sequence: 0, 1, 2, 6, 18, 45, 102, 213, 426, 821, ... [ OEIS ]

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Is the 6-polyomino Snaky achievable (by player 1) on an infinite grid, or can it be blocked by player 2? Combinatorial games [ wiki ] What is the smallest grid on which Snaky can be achieved by player 1? [ jstor ] (Gardner)

Specific Question: Can Snaky win on a 9 x 9 board? Best known answer: Snaky is a loser on an 8 x 8 board. [ Research Gate ]

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What is the largest number of times any number > 1 can appear in Pascal's triangle? Singmaster's conjecture [ wiki ] What number n > 1 shows up as a combination more times than any other number?

Specific Question: Are there any entries of Pascal's triangle that appear more than 8 times? Best known answer: 3003 appears 8 times. (Unknown if any others appear 8 times.) Sequence: 2, 3, 6, 10, 120, 120, 3003, 3003, ... OPEN [ OEIS ]

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How many n x n matrices are there with all entries 0 or 1, and with nonzero determinant? How many n x n invertible (nonsingular) matrices are there with all entries 0 or 1? [ Mathworld ]

Specific Question: How many 9 x 9 invertible matrices are there with all entries 0 or 1? Best known answer: There are 10160459763342013440 invertible 8 x 8 matrices with all entries 0 or 1. Sequence: 1, 1, 6, 174, 22560, 12514320, ... OPEN [ OEIS ] Bonus: what is the convex hull of these n x n matrices?

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What is the largest determinant of an n x n matrix with all entries 0 or 1? [ wiki ]

Specific Question: What is the largest determinant of an 22 x 22 matrix with all entries 0 or 1? Best known answer: The largest determinant of a 21 x 21 size (0,1)-matrix is 195312500. Sequence: 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, ... OPEN [ OEIS ]

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• Does every finite connected vertex-transitive graph contain a Hamiltonian cycle except the five known counterexamples?
• Is Catalan's constant irrational?
• How many finite topologies are there, with n elements?
• How many graphs are there up to isomorphism on n nodes?
• How many transitive relations are there on a set with n elements?
• How many distinct combinatorial polytopes are there, in 3D, with n vertices?
• Is every even number >0 the sum of two primes?
• Do two knots with n, m respective minimum crossings connect-sum to a knot with n+m crossing number? (Compare: knot checkerboard coloring and strand diagrams!)
• Can the triple connected sum of some knot (3 copies) be cobordant to the unknot? slice? ribbon?
• Is every self-inverse-concordant knot (having 2-torsion) amphichiral?
• Are the nontrivial zeroes of the Riemann zeta function all on the line Re(z) =1/2?
• Is the diameter of a polytope always found by a polynomial using its number of facets and dimension? (Polynomial Hersch)
• What are the homotopy groups of spheres? Stable homotopy groups of spheres? The ways to map a 70-dimensional sphere onto a 2-sphere?
• What is the value of the busy beaver function for n=6? That is, what is the maximal number of steps that an n-state, 2-symbol, d+ in {LEFT, RIGHT}, 5-tuple (q, s, q+, s+, d+) Turing machine can make on an initially blank tape and then halt?
• Can a thrackle have more edges than vertices?
• Does every convex polyhedron have a net?
• How many nets does the n-dimensional hypercube have?
• Is the Mandlebrot set path-connected?
• Are multiassociahedra polytopes? (Does Loday's realization generalize?)
• Are pseudograph-multiplihedra polytopes? Poset multiplihedra?
• Does the Collatz sequence always end in 1?
• Does P = NP? Is there a polynomial time method to decide if a logical expression is a contradiction or not?
• Are there more than 5 Fermat primes? (more than 31 constructible odd-sided regular polygons?)
• Is pi + e irrational?
• Does every finite sequence of numbers appear in pi?
• Is pi normal? e?
• Is the (computable) omega constant normal? (This omega is the solution to xe^x=1.)
• What is the sixth bit of the incomputable (normal) Omega corresponding to the Busy Beaver problem? (This omega is Chaitin's constant.)
• What is the input n for the busy beaver value that would determine Goldbach?
• Is there a finite projective plane of order 12?
• How many unique (up to constant factor) non-negative solutions to n-1 homogeneous linearly independent linear equations in n variables using 1, 0 ,-1 as coefficients are there?
• How many oriented matroids are there on a base set of size n? Up to isomorphism?
• How many incidence geometries are there for n points? n lines? Up to isomorphism?

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