a) In how many ways can the word M A T E M A T I K A be read from the diagram (by stepping to a neighboring letter from each letter), and in how many ways can the word M E T A M A T E M A T I K A be read?

b) By omitting some letters, Feri modified the diagram so that the word M A T E M A T I K A could be read in exactly 500 ways. Find such an omission instruction.

![](https://cdn.mathpix.com/cropped/2024_05_02_35c5a6a02c62b5117a2bg-1.jpg?height=683&width=702&top_left_y=347&top_left_x=668)
![](https://cdn.mathpix.com/cropped/2024_05_02_644936b5e2c3be9590f6g-1.jpg?height=740&width=680&top_left_y=126&top_left_x=709)

Figure 1

a) The reading of the word MATEMATIKA can obviously only start from the $M$ at the center of symmetry of the diagram and end at one of the $A$ letters in the corners. Due to symmetry, it is sufficient to count the number of reading paths ending at the $A$ in the lower right corner, and the desired number is four times this. Thus, we always have to move either to the right or downward, and we can only step from the left or upper neighbor of each letter - for letters on the axes and the $I$ letters next to the dark square, only from the single such neighbor. Therefore, we can reach each letter in as many ways as the sum of the ways to reach its two neighbors, or the single neighbor. These numbers are shown in Figure 2, with 66 paths leading to each terminal letter, so the word MATEMATIKA can be read in $8 \cdot 66 = 528$ ways from the diagram.

![](https://cdn.mathpix.com/cropped/2024_05_02_644936b5e2c3be9590f6g-1.jpg?height=691&width=553&top_left_y=1313&top_left_x=775)

Figure 3

b) The word METAMATEMATIKA, however, cannot start at the center, as the $M$ there has no $E$ neighbor, but it can start at any other $M$; furthermore, the 5th letter must again be the central $M$. Thus, it is sufficient to focus on the sequence $M E T A M$, and each of its reading possibilities can be completed in 528 ways. The solution is given by Figure 3, with $60 \cdot 528 = 31680$ reading possibilities. (The problem's requirements do not exclude, for example, the same $A$ appearing as the 6th letter as it did as the 4th.)

c) To reduce the number of reading ways of the word MATEMATIKA to 500, we first determine for each letter - but again only in the lower right part of the diagram - how many reading paths pass through it. Indeed, by omitting only that letter, the number of reading possibilities decreases by this amount. This - similar to the connection between METAM and MATEMATIKA - is given by the product of the number of ways to reach the letter from the starting letter and the number of ways to reach any terminal letter from the letter. The latter number is the same as the number of paths leading from the two terminal letters to the letter, as if reading the word backward.

![](https://cdn.mathpix.com/cropped/2024_05_02_644936b5e2c3be9590f6g-2.jpg?height=382&width=358&top_left_y=159&top_left_x=886)

Figure 4

These numbers are shown in Figure 4, and the product of the numbers in the 2nd and 4th figures for each cell is shown in Figure 5.

![](https://cdn.mathpix.com/cropped/2024_05_02_644936b5e2c3be9590f6g-2.jpg?height=401&width=353&top_left_y=762&top_left_x=867)

Figure 5

However, we did not write down the products greater than $528 - 500 = 28$ (an $x$ symbol is used instead), as omitting the corresponding letter would reduce the number of readings below 500.

In Figure 5, we wrote a factor of 2 before the 3 products in the two symmetry axes of the original diagram, indicating that omitting the corresponding letter would also eliminate the same number of readings in the upper right and lower left quarters of the diagram.

There is no 28 in Figure 5, so reducing to 500 cannot be achieved by omitting a single letter. However, if we try to omit more letters, the reduction is only given by the sum of the numbers found for them if there is no reading path that passes through more than one of the letters to be omitted.

The reduction of 28 can be achieved in several ways from the found numbers:

$$
2 + 2 + 6 + 6 + 6 + 6, \quad 2 + 2 + 6 + 6 + 12, \quad 2 + 2 + 12 + 12
$$

and each of these can be achieved in several ways by omitting 6, 5, or 4 letters. The 6th figure shows sketches of such omissions, also omitting letters that become unreachable (dead-end) due to the omission.
![](https://cdn.mathpix.com/cropped/2024_05_02_644936b5e2c3be9590f6g-2.jpg?height=418&width=1140&top_left_y=1840&top_left_x=488)

Figure 6

This solution was printed before the submission of the solutions.