# The most profitable way to plant potatoes

Many of us have our own garden, or often go to the country, where they grow potatoes and other vegetables. In order to rationally use the area of the garden and get the maximum yield, you need to learn how to plant vegetables properly. For example, in order to get the maximum yield of **potatoes, their planting should be done at some equal intervals**, as prescribed by science, based on many years of experience. The question is how best to place the holes on the field. And here not only agronomists, but also mathematicians can say their word.

As you know, there are three polygons into which you can divide the area without spaces or breaks: an isosceles triangle, a square and a regular hexagon. So, only these three types of relative placement of potato holes can be taken into account. When using a hexagon, the earth will not be (due to the necessary constant distance between the holes) fully utilized, it is almost obvious. Doubts can arise only when choosing a square or triangle. We place one plant in the center of each of these polygons, and we select the sizes of the polygons so that the distance between the closest plants is the prescribed size, for example d = 56 cm. (If someone wants to do calculations for a different value of d, then this will not be more complicated than in this case we are analyzing.)

When planting potatoes in a square, as shown in Figure 1, we must place rows at a distance of 56 cm and plant potatoes in each row at a distance of 56 cm from each other. Then each plant will have 56 * 56 = 3136 cm^{2} of soil, and it will be possible to place 1000000: 3136 = 319 holes in one arena (10x10 m).

We will also analyze the method of planting potatoes at the tops of isosceles triangles (Fig. 2).

With this method of planting, each plant will have a regular hexagon in which the distance from the center to the faces will be 28 cm. This regular hexagon can be divided into 6 isosceles triangles.

The height of each of these triangles will be 28 cm, and the side, as it is easy to determine, about 32 cm. The area of such a small triangle will be 0,5 * 32 * 28 = 448 cm^{2}, and the entire regular hexagon will take 6 * 448 = 2688 cm^{2}. On one arena, you can place 1000000 : 2688 = 372 holes of potatoes, that is, more than when planted in a square.