Advice 1: How to draw a circle with a dot, without lifting the pencil

In forums often for the purpose of trolling the question about how to draw a circle with a dot without lifting the pencil from the paper. In fact, this problem has many solutions. Most of them are based on inaccurate formulation of the problem.
How to draw a circle with a dot, without lifting the pencil
In the problem does not specify where exactly should be the point. Draw it right on the circle and not inside it - and a formal decision is made.
Try the inaccuracy in terminology. The condition uses the term "circle" instead of "circumference". Unlike the circle, the circle is solid. Draw it, then put inside it the bullet. This can be done without lifting your pencil from the paper.
In the problem statement does not say whether to use the second pencil. Also not specified how many pencils can not be separated from the paper. Draw a dot inside the second circle with a pencil, without lifting off paper first.
The problem statement does not contain information about whether the circumference to connect to a point inside the line. Assuming that the drawing of such a line is not prohibited, draw a circle, and then, without lifting your pencil from the paper, draw a line inside it, and then draw a point. Moreover, because the condition does not say definitely whether the point should be inside the circle, try to draw it outside.
Being a true formalist, to solve this problem. Since in the condition they say it is the pencil, not the pen, the pen, press it onto the paper, the pencil, and then, without lifting it, draw another drawing tool, a circle and a point inside it.
The condition is not stated which part of the pencil can not be separated from the paper. Press to the sheet (same or different - in condition nothing is said on this subject) the opposite side of the pencil and the pen, draw a circle and a point with margin.
Draw a circle with a compass. Dot in its center will turn itself. Since the compass includes a pencil, formally, the task is considered solved.
Finally, the most elegant way of solving this problem is the following. Draw a circle, then fold the corner of the paper along with a pencil, without lifting it, so that the stylus touched the paper with the reverse side in the center of the circle. Then pierce the paper with a stylus.
If forum trolls will tell you in response that none of the proposed solutions to the problem is not correct, please provide your counter. It is as follows: to solve the problem is impossible, because after a circle with a dot is drawn (regardless of how), the pencil will have the paper to tear, and the condition of the problem forbids it. Do not glue it. But better to remember the famous advice: "don't feed the trolls".

Advice 2: How to draw a circle and a dot in the center without lifting your pencil

The circle and dot in the center is one of the oldest mathematical problems, the solution of which is essentially for many Buddhist enlightenment sort of cotton one hand. The reason of this task is to teach the subject to break away from the framework of standard thinking in strictly specified directories and to force to think in more than two axes of the coordinate system.
How to draw a circle and a dot in the center without lifting your pencil
You will need
  • pencil;
  • - a sheet of paper.
Carefully review the terms of the task assigned to you. Note the points about the drawing surface, the possibility of changing the surface and work in two-dimensional space. If a task has such a clause (e.g., "draw a circle and put a dot, using only two-dimensional space"), such a problem has no solution.
Take a sheet of loose paper. It is necessary, that it could well be bent, retaining traces of folds. Using a pencil, draw on paper a circle so that its edges almost touch the edge of the sheet. Since the task does not require strict adherence to geometric shapes, the circle may not be perfect. It is worth remembering that after the beginning of the tracings of the circle to tear the pencil from the paper impossible.
Fold the paper so that it touched the opposite edge, and repeat the operation. It is worth Recalling that tear the pencil from the line of the circle is impossible. Bend the sheet in the opposite to the pattern side out. In the end, a drawn circle and the crease line sheet form some semblance of a target – a circle and divide it into four equal parts cross.
Finally, fold the paper so that the edge of the circle with recorded with a pencil touched the center of the cross – points of intersection of bend lines. The problem is solved: the point in the center of the circle is set, and the pencil was torn from the circle and the whole worksheet. It should be noted that some teachers think this decision is invalid, in this case it is necessary to re-bend the leaf out and put the dot in the center of the circle using the puncture the surface of the sheet.
Such tasks are designed to develop logical thinking of novice mathematicians. This task, in particular based on the departure from the two-dimensional coordinate system (length and height) and working in three-dimensional space (length, height and depth). Some tasks require more care in the four-dimensional universe, where for solving the problem will need to take into account the time.
Is the advice useful?