The probability that 3 cards drawn at random from the Set deck—the game we have focused on for the past two weeks—form a set is simply the probability that the third card drawn is the only one out of the 79 remaining that can be “joined” with the first two, whatever they may be, that is, 1/79.
Since the game begins by placing 12 cards at random on the table, there will be as many possible starts as there are combinations of 81 elements taken 12 by 12, that is, 81.80.79…70/12!: about 70.7 billion.
The maximum number of cards that can be on the table without any set being formed is 20; although with 19 the probability of that happening is already less than one in a billion. As a curious anecdote, this was demonstrated by the Italian mathematician Giuseppe Pellegrino in 1971before the Set was created (but that’s another article).
And regarding the average number of sets that we can expect with 12 cards on the table, this is what our “featured user” Francisco Montesinos comments: “If we have 12 cards drawn at random, to form a set the first one can be chosen from 12 ways, the second out of 11 and the third has a probability p = 10/79 of being in the remaining ten, so the probable number of sets will be 12.11.(10/79)/6 = 220/79 = 2, 78″.
In the case of the reduced Set, with only 3 characteristics and 27 cards, the probability that 3 randomly drawn cards will form a set will be 1/25, since, for any initial pair, only one of the remaining 25 cards will form a set with them two.
No more wax than burns
With the paschal candles still smoldering, it seems like a good time to raise some issues regarding candles and their burning times:
1. On Monday I lit a candle and kept it burning for an hour. On Tuesday I lit two candles and kept them burning for an hour. On Wednesday I lit three candles and kept them burning for an hour, and so on, lighting one more candle each day and keeping them burning for an hour… until today, when I have run out of candles. Knowing that each candle takes 4 hours to completely burn out, what day is it today?
2. “How fast time flies! —Says an old man sadly as he puts away the candles on his last anniversary cake. The day before yesterday there were 77 candles on my birthday cake and next year I will need 3 more”. When is his birthday?
3. We have two cylindrical candles of the same height and the same material, but one slightly thicker than the other, so that one takes 4 hours to burn completely and the other takes 5 hours. At one point we extinguish both of them at the same time and we see that the height of the thickest candle is four times greater than that of the other. How long has it been?
And, finally, one very similar to the previous one taken from the XX Mathematical Olympiad:
4. We have placed two candles of different heights in the garden. The longest measures 28 cm and takes 7 hours to burn, while the shorter one, which is thicker, takes 11 hours to burn. We turn them both on at the same time when the party starts and after 3 hours, when the friends leave, we turn them off. At that moment they are both the same height. What height was the shortest sail originally?
You can follow SUBJECT in Facebook, Twitter and instagramor sign up here to receive our weekly newsletter.
#enigmatic #candles