Reflections on the way to school. 1. On foot

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REFLECTIONS ON THE WAY TO SCHOOL

1. On foot

The sum of house numbers on a block (a section of street between two intersections) is 33. What numbers are these?

It is easy to report the following:

1. The houses are located on the odd side, since the sum of four numbers is always.

2. The number of houses on a block is odd because the sum of four odd numbers is also even.

3. If the number of houses is odd, then there is an average house. Let his number be $l$. Neighboring houses have numbers $l+2$ and $l-2$, neighboring ones $l+4$ and $l-4$, etc. Thus, the sum of the numbers equidistant from the average is 2$l$ and, therefore, the double sum of house numbers on the block is 2$lm$, where $m$ is the number of houses. This means that the sum of the house numbers on the block is $lm$ and is not a prime number.

In our case, the number 33 can be decomposed into two copartners in two ways. 1) 33=3×11, 2) 33=1×33 It is clear that $m<l$ since negative numbers are not assigned to houses. This means that either $l=11$ and the block consists of houses numbered 9, 11, 13, or $l=33$ and the whole block consists of one house 33.

Solve the same problem if the sum of the house numbers is 333.

A. Chulsky

(Continued on page 59)