Balti Tee 1991 · Ülesanne 15
Kombinatoorika
In each of the squares of a chess board an arbitrary integer is written. A king starts to move on the board. As the king moves, 1 is added to the number in each square it "visits". Is it always possible to make the numbers on the chess board:
(a) all even;
(b) all divisible by 3 ;
(c) all equal?
Kui oled valmis
Ülevaatematerjal muutub kättesaadavaks järgmise päevaülesannete komplektiga.
Ülevaade
Teemad
Mängud ja strateegiad · Värvimised ja konfiguratsioonid
Lahendused
Lahendus
Figure 1
Figure 1 demonstrates a possible king's path passing through each square exactly once and finally returning to the initial square. Thus, it suffices to prove part (c) as we can always increase the numbers in all the squares by 1 or 2 if necessary. Moreover, note that for any given square it is possible to modify the path shown in Figure 1 in such a way that this particular square will be passed twice while any other square will still be passed exactly once. Repeating this procedure a suitable number of times for each square we can make all the numbers on the chess board equal.