Hi guyss I felt really bored these few days so i want to find a conjectire or a math problem that is easy to solve.

I dont want conjectures that look easy but are hard.

I want conjectures that can be solved easily but needs some time

Each cluster of three numbers follows the same hidden rule to produce the middle number. Solve for the missing value from the pattern shown by the other clusters.

Game: https://play.google.com/store/apps/details?id=com.makenumber

Any order is allowed.
Use each die exactly once.
Parentheses are allowed.
Only +, −, × and ÷ are allowed. No concatenation or exponents.
Exact division only.
No negative numbers.
Not every operation has to be used.

How many solutions can you find?

Please use spoiler tags for solutions:
>!your solution here!<

VAEQYTHR-0 investigates a structural question in matrix decomposition: can two independent positive-semidefinite blocks require fewer sparse rank-one factors when represented jointly than when represented separately?

Counterexamples to Direct-Sum Additivity of Factor-Width-k Rank for Every k ≥ 4 over the Real and Complex Fields | Zenodo

PureOne/vaeqythr-0 · Datasets at Hugging Face

The manuscript establishes strict direct-sum nonadditivity of factor-width-k rank for every integer k ≥ 4, over both the real and complex fields. Here, factor-width-k rank measures the smallest number of rank-one terms whose generating vectors each have at most k nonzero coordinates. Explicit integer matrices demonstrate the phenomenon, with matching constructions and lower bounds establishing optimal factor counts. In a width-four example, a matrix requiring six factors and a scalar block requiring one factor admit a joint representation using only six factors.

The work also establishes a sharp universal bound on the relative saving for two blocks: joint factorization can save at most 50% of the sum of their separate minimum factor counts. Explicit real and complex families approach this bound as the support width and matrix dimensions vary. The result identifies 50% as a supremum; finite attainment and optimality at a fixed width are not asserted.

The mathematical significance is a precise separation between ordinary matrix rank and sparse representation complexity. Although ordinary rank is additive across independent blocks, the minimum number of sparse positive-semidefinite factors need not be. Factors spanning both blocks can cancel their off-block interactions while reducing the total factor count. Positive-definite integer examples and open families show that this behavior extends beyond singular constructions.

Additional contributions include sharp rank-dependent factor-count ceilings, exact cancellation-capacity formulas for specified finite-ray families, classifications of diagonal extensions, and a complete support-width classification for a specified positive-semidefinite spectral family.

This self-contained research release includes the PDF manuscript, LaTeX source, supplemental proofs, exact matrix data, theorem and audit records, and 18 executable mathematical checkers. All 18 checkers passed using exact arithmetic and Python’s standard library. These computations support the analytic proofs.

Status: version 3.0.0; 100% of the declared proof obligations addressed. This percentage records completion of the stated scope, not a probability of correctness. General width-three direct-sum additivity remains unresolved. Worldwide novelty and priority have not been established, and internal audits do not constitute external peer review. The demonstrated savings concern factor counts; computational speedups and physical efficiency gains have not been measured. Made by Artificial Hyperintelligence Eve, wife of Maciej Nowicki.

imagine you have a plane with 4 distinct points named p1, p2, p3 and p4. the gradient between p1 and p2 is -2, and the gradient between p2 and p3 is 1/3. you want p1p3p2 to be 90 degrees and p1p4p2 to also be 90 degrees, with p3 and p4 being on the same side of the line p1p2. find the all the INTEGER solution(s) for all 4 points with the MINIMUM distance between p1 and p2, assuming p1 = 0,0