Intereting Posts

How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
$M_n\cong\Gamma(\operatorname{Proj}S.,\widetilde{M(n).})$ for sufficiently large $n$
a conjecture of certain q-continued fractions
Every real sequence is the derivative sequence of some function
When is $\sin(x)$ rational?
Can one avoid AC in the proof that in Noetherian rings there is a maximal element for each set?
Pointwise a.e. convergence implies strong convergence?
How to compute the pade approximation?
Surjective function into Hartogs number of a set
How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?
Is the polynomial $x^{105} – 9$ reducible over $\mathbb{Z}$?
No husband can sit next to his wife in this probability question
If $(\nabla f(x)-\nabla f(y))\cdot(x-y)\geq m(x-y)\cdot(x-y)$, why is $f$ convex?
Show that $\lim_{n\to\infty}n+n^2 \log\left(\frac{n}{n+1}\right)= 1/2$
Show that if $ar + bs = 1$ for some $r$ and $s$ then $a$ and $b$ are relatively prime

Sometimes I help my next door neighbor’s daughter with her homework. Today she had to trisect a line segment using a compass and straightedge. Admittedly, I had to look this up on the internet, and I found this helpful site. There they claim that the fewest number of “steps” necessary to trisect the segment is $4$, where by one “step” we mean any stroke of the pencil, either with the compass or straightedge.

Immediately, this got me thinking about the length of other optimal constructions, which has led to the question of this post:

What is a the minimum number of steps necessary to construct a segment of length $\frac{1}{n}$ given a segment of length $1$?

- Drawing a triangle from medians
- Construct parallel line equidistant from two other parallel lines - Euclidea on iOS challenge 2.8
- Compass-and-straightedge construction of the square root of a given line?
- How to construct a line with only a short ruler
- Construction using a straight edge only
- Can a circle's circumference be divided into arbitrary number of equal parts using straight edge and compass only?

If $s(n)$ is the quantity in question, then this other helpful site shows that $s(n)\le n+6$. However, $s(2)=3$ and $s(3)=4$, so the bound is not sharp. Also, we can see that $s(mn)\le s(m)+s(n)$ by creating a segment of length $\frac{1}{mn}$ from one of length $\frac{1}{n}$.

Finally, at the bottom of the first site, they hint at one method of construction, which involves drawing larger and larger circles. Assuming their hint leads to an optimal construction (which would need to be proved), I believe that the first eight values of $s(n)$ starting with $n=1$ are:

$$0,3,4,5,5,5,5,6$$

This returns nothing on OEIS. (The above numbers assume that the initial segment of length $1$ is marked off on a very long ray, else we’d have to add one for $n\ge3$ to lengthen the segment appropriately).

Any ideas?

- A variant of assignment problem (different sizes of sets)
- Orthogonal Projection onto the $ {L}_{1} $ Unit Ball
- Quasiconvex and lower semicontinuous function
- Variable leaving basis in linear programming - when does it happen?
- Maximum and minimum of an integral under integral constraints.
- If $A+B+C+D+E = 540^\circ$ what is $\min (\cos A+\cos B+\cos C+\cos D+\cos E)$?
- Best fit ellipsoid
- Confounding Lagrange multiplier problem
- How is the Lagrangian related to the perturbation function?
- How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac n2 \rceil$ or $ \lfloor \frac n2\rfloor $?

Let starting segment is $AB$.

As one can see from first link, starting condition is “segment-on-the-line”.

Anyway one can add $1$ line at the start to get this starting condition.

Consider **odd** $n$.

Let coordinates of starting points are: $A(-1,0)$, $B(0,0)$.

If point $C$ has coordinates $C(n,0)$, then (see **Figure 1**) coordinates of other points are:

$D\left(\dfrac{1}{2n},\dfrac{\sqrt{4n^2-1}}{2n}\right)$;

$\qquad$

$E(1,0)$;

$\qquad$

$P\left(-1+\dfrac{1}{n},0\right)$.

**Figure 1**:

Consider **even** $n$.

Let coordinates of starting points are: $A(0,0)$, $B(1,0)$.

If point $C$ has coordinates $C\left(\dfrac{n}{2},0\right)$, then (see **Figure 2**) coordinates of other points are:

$D\left(\dfrac{1}{n},\dfrac{\sqrt{n^2-1}}{n}\right)$;

$\qquad$

$E\left(\dfrac{1}{n},-\dfrac{\sqrt{n^2-1}}{n}\right)$;

$\qquad$

$P\left(\dfrac{1}{n},0\right)$.

**Figure 2**:

**Main idea**: For given $m$ to draw point $C(m,0)$ as fast as possible.

As I checked (up to $m=2048$), it is possible to draw point $C(m,0)$, applying

$$

1+\lfloor \log_2 m \rfloor

$$

steps, where $\lfloor \cdot \rfloor$ is floor rounding function.

According to described construction, upper bound of (total) steps is

$$

3+\lfloor \log_2 n \rfloor, \mbox{ if } n \mbox{ is odd} ;

$$

$$

2+\lfloor \log_2 n \rfloor, \mbox{ if } n \mbox{ is even}.

$$

Upper bounds of steps (starting with $n=1$) are:

$$

0; ~~ 3, 4; ~~ 4, 5, 4, 5; ~~ 5,6,5,6,5,6,5,6; ~~ 6,7,6,7, …

$$

Examples:

$n=11$: build point $C(11,0)$ and follow figure 1 (total $6$ steps).

$n=12$: build point $C(6,0)$ and follow figure 2 (total $5$ steps).

- Combinatorial proof involving partitions and generating functions
- Quotient of the ring of integers of a number field by a prime ideal
- Finding the Gradient from these functions.
- Trying to show Tautology
- A die and a coin are tossed one after the other
- Examples of problems that are easier in the infinite case than in the finite case.
- What is the real life use of hyperbola?
- Is it possible to calculate this integral
- Combinatorics/Task Dependency
- How to prove that derivatives have the Intermediate Value Property
- Closed formula for the sum of the following series
- Determine if a point is inside a subtriangle by its barycentric coordinates
- Finding a choice function without the choice axiom
- Property of Derivative in a local field
- how many unique patterns exist for a NxN grid