Open mathematical problems

Antihydra is a sequence starting at 8, and iterating the function $H(n) = \left\lfloor \frac {3n}2 \right\rfloor.$ The conjecture states that the cumulative number of odd values in this sequence is never more than twice the cumulative number of even values. It is a relatively new open problem with, so it might be solvable, although seems quite hard because of its Collatz-like flavor. The underlying Collatz-like map has been studied independently in the past, see doi:10.1017/S0017089508004655 (Corollary 4).

It is equivalent to non-termination of the 1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA 6-state Turing machine (from all-0 tape). Note that the conjecture that the machine does not halt is based on a probabilistic argument.

This machine and its mathematical reformulations were found by bbchallenge.org contributors mxdys and Rachel Hunter on June 28th 2024.

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openOther · Combinatorics

Beaver Math Olympiad (BMO): Set

BMO#2 formulation variant

Alternative statement of beaver_math_olympiad_problem_2_antihydra using set size comparison instead of a recurrent sequence b.

Source checked Jul 26, 20261 pinned Lean statementInspect problem

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Beaver Math Olympiad (BMO)

BMO#5

Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be two sequences such that $(a_1, b_1) = (0, 5)$ and

(a_n+1, b_n-f(a_n)) & \text{if } b_n \ge f(a_n) \\ (a_n, 3b_n+a_n+5) & \text{if } b_n < f(a_n) \end{cases}$$ where $f(x)=10\cdot 2^x-1$ for all non-negative integers $x$. Does there exist a positive integer $i$ such that $b_i = f(a_i)-1$? [BMO#5](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#5._1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE_(bbch)) is equivalent to asking whether the 6-state Turing machine [`1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE`](https://wiki.bbchallenge.org/wiki/1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE) halts or not. There is presently no consensus on whether the machine halts or not, hence the problem is formulated using `answer(sorry) ↔`. The machine was discovered by [bbchallenge.org](bbchallenge.org) contributor mxdys on August 7th 2024. The correspondence between the machine's halting problem and the below reformulation has been proven in [Rocq](https://github.com/ccz181078/busycoq/blob/BB6/verify/1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE.v).

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openOther · Combinatorics

Beaver Math Olympiad (BMO)

BMO#8

Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be two sequences such that $(a_1, b_1) = (10, 12)$ and

(a_n - \lfloor b_n/2 \rfloor - 3, 3 \lfloor (b_n+1)/2 \rfloor + 6) & \text{if } a_n > \lfloor b_n/2 \rfloor \\ (3 a_n + 5, b_n - 2 a_n) & \text{if } a_n \le \lfloor b_n/2 \rfloor \end{cases}$$ for all positive integers $n$. Does there exist a positive integer $i$ such that $a_i = \lfloor b_i/2 \rfloor + 1$? [BMO#8](https://wiki.bbchallenge.org/wiki/Beaver_Math_Olympiad#8._1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA_(bbch)) is equivalent to asking whether the 6-state Turing machine [`1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA`](https://wiki.bbchallenge.org/wiki/1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA) halts or not. There is presently no consensus on whether the machine halts or not, hence the problem is formulated using `answer(sorry) ↔`.

Source checked Jul 26, 20261 pinned Lean statementInspect problem

Source labels openOther · Combinatorics

VCₙ dimension of convex sets in ℝⁿ, ℝⁿ⁺¹, ℝⁿ⁺²

Every convex set in $\mathbb R^3$ has $\mathrm{VC}_2$ dimension at most 1.

Source checked Jul 26, 20261 pinned Lean statementInspect problem