[ $\left[\begin{array}{ll}\text { Decimal number system } & \\ \text { Evenness and oddness } & ]\end{array}\right]$ [Pairing and grouping; bijections ]

How many more natural numbers from 1 to 1000000 (inclusive) are there: even numbers with an odd sum of digits or odd numbers with an even sum of digits?
Let's call the numbers mentioned in the problem statement interesting. First, consider the first two dozens. Among the numbers from 1 to 9, there are no interesting numbers, while all numbers from 10 to 19 are interesting, and among them, there are 5 even numbers with an odd sum of digits and 5 odd numbers with an even sum of digits.

Now, consider any sequence of ten consecutive numbers, from *0 to *9 (the symbol * represents the same sequence of digits). If the sum of the digits in the sequence * is even, then there will be no interesting numbers in this decade. If the sum is odd, then all numbers in this decade will be interesting, and there will be an equal number of each type. When dividing the number sequence from 10 to 999999 into such decades, there will be an equal number of interesting numbers of both types, but the number 1000000 will remain, which, being even, has an odd sum of digits, so there will be more of such numbers.

## Answer

Even numbers with an odd sum of digits.

![](https://cdn.mathpix.com/cropped/2024_05_06_c07b2c4be0d3dcbcc4e5g-23.jpg?height=91&width=1512&top_left_y=26&top_left_x=14)
[ Examples and counterexamples. Constructions ]

Author: $\underline{\text { Folkiyor }}$

There are 40 identical cords. If any cord is lit from one end, it burns, but if lit from the other end, it does not burn. Vasya arranges the cords in the form of a square (see the figure, each cord is a side of a cell). Then Petya places 12 fuses. Can Vasya arrange the cords in such a way that Petya cannot burn all the cords?

![](https://cdn.mathpix.com/cropped/2024_05_06_c07b2c4be0d3dcbcc4e5g-23.jpg?height=280&width=278&top_left_y=383&top_left_x=890)

## Solution

We will provide a construction where 12 fuses are insufficient. We can assume that Petya is allowed to place fuses only at the ends of the cords. After all, if a fuse is moved from an internal point of a cord to the end from which it burns, the situation does not worsen.

Place an arrow on each cord in the direction it can burn. Color the nodes of the grid in a checkerboard pattern. There will be thirteen nodes of one color, let's say black. Assume that all arrows point out from these black nodes (see the figure). Then each cord that exits from a black node can burn only if a fuse is placed in that node. But there are only 12 fuses, and 13 nodes.

![](https://cdn.mathpix.com/cropped/2024_05_06_c07b2c4be0d3dcbcc4e5g-23.jpg?height=463&width=460&top_left_y=1118&top_left_x=799)

## Answer

Yes, he can.