34. The network of irrigation ditches represents a "branched" broken line (Fig. 87) passing through the field; in this case, the considered broken line is "connected," i.e., consists of one piece, since all ditches are connected to the source J. Let us single out from the system of ditches some "main ditch," i.e., a "simple" (unbranched) broken line $A B$. If the network of ditches has another end $C$ different from $A$ and $B$, consider the "additional ditch" going from $C$, i.e., a simple broken line $C C^{\prime}$, which at some point $C^{\prime}$ (possibly coinciding with the source $J$) flows into the main ditch $A B$; further, if the broken line has another end $D$ different from $A, B$, and $C$, then consider the "additional ditch" $D D^{\prime}$ leading from $D$, which at some point $D^{\prime}$ flows into the system of ditches $A B$ and $C C^{\prime}$, and so on.

Now let us consider what the "irrigated area" represents, i.e., the set of points located no more than 1 unit away from the nearest ditch. For a straight ditch $A B$ of length $l$, the irrigated area is a rectangular strip $A_{1} B_{1} B_{2} A_{2}$ of length $l$ and width 2, supplemented by two semicircles of radius 1 (Fig. 88); the area of the corresponding figure is $2 l + \pi$. If two straight ditches $A M$ and $M B$ meet at point $M$, forming a broken line $A M B$ (Fig. 89, a, b), then the rectangles $A_{1} M_{1} M_{2} A_{2}$ and $M_{1}^{\prime} B_{1} B_{2} M_{2}^{\prime}$ corresponding to segments $A M$ and $M B$ are supplemented by a circular sector $M_{2} M M_{2}^{\prime}$ of radius 1, with a central angle of $180^{\circ} - \angle A M B$; however, the centrally symmetric sector $M_{1} M M_{1}^{\prime}$ relative to point $M$ (and not only it!) is covered twice, so the area irrigated by the "broken ditch" $A M B$ does not exceed $2 A M + 2 M B + \pi = 2 l + \pi$, where $l$ is the length of the broken line $A M B$. Similarly, it is shown that for a "broken ditch" $A M N \ldots R B$ with any number of "internal vertices" $M, N, \ldots, R$, the irrigated area (Fig. 90) does not exceed $2 l + \pi$, where $l$ is the length of the broken line $A M N \ldots R B$.

Now consider a straight ditch $C C^{\prime}$ of length $m$; "flowing into" the ditch $A B$ at point $C^{\prime}$ (Fig. 91, a). The addition of the ditch $C C^{\prime}$ to the ditch $A B$ increases the area irrigated by the ditch $A B$ by (overlapping with the area irrigated by the ditch $A B!$) the rectangle $C_{1} C_{1}^{\prime} C_{2}^{\prime} C_{2}$ and the semicircle $C_{1} C_{2}$ of radius 1 centered at $C$. However, the shaded semicircle $C_{1}^{\prime} C_{2}^{\prime}$ centered at $C^{\prime}$ (and not only it!) is covered twice, so the addition of the ditch $C C^{\prime}$ of length $m$ to the ditch $A B$ increases the irrigated area by no more than $2 m$. This conclusion remains valid even if the straight ditches $A B$ and $C C^{\prime}$ are replaced by broken ditches $A M N \ldots R B$ and $C S T \ldots U C^{\prime}$, the total length of the latter being $m$ (Fig. 91, b).

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a)

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Fig. 91.

From the above, it follows that if the total length of the ditches is $L$, then the irrigated area does not exceed $2 L + \pi$. In particular, if the total length of the ditches does not exceed 70, then the irrigated area does not exceed

$$2 \cdot 70 + \pi < 140 + 3.15 < 144 = 12^{2}$$

from which it follows that it cannot coincide with the entire field.

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a)

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Fig. 92.