Help figuring a formula for my job

Solution 1:

This is a very well known, difficult problem, called the "bin packing problem". There is no known algorithm that would work efficiently for large numbers of pipes, but for reasonably small numbers it can be done in practice.

A reasonably good, though not always optimal, strategy is "first fit decreasing". Sort the desired output lengths in decreasing order, and cut each from the first tube where it will fit.

Hausdorff spaces from filters

Question 43 in section 2.2 of Hatcher's Algebraic Topology

Is the following always prime?

Can a sequence converges modulo every r>0 but diverge?

Evaluate $\int _0^1\int _0^1\int _0^1\int _0^1\sqrt{(z-w)^2+(x-y)^2} \, dw \, dz \, dy \, dx$

For a differentiable function $f$ show that $\{x:\limsup_{y\to x}|f'(y)|<\infty\} $ is open and dense in $\mathbb R$

Bank vaults and probability

Old USAMO combinatorics problem about distribution of members into committees containing a fixed number of members.

Sup norm of Fourier transform of $ \frac{\sin |x|}{|x|^\lambda} \mathbb 1_{\{2^k\le |x| <2^{k+1}\}}, \ 0<\lambda<n $

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How to show $\lim_{n\to\infty}n\cdot \sum_{m=1}^{\infty}\Big(1-\frac{1}{m}\Big)^n\cdot \frac{1}{m^2}=1.$

Show that a group of order $pq$ has subgroup of order $p$ and $q$ without using sylow's and cauchy's theorem