# Mysterious Candles | flag game

The probability of drawing 3 cards at random from the deck – the game we’ve focused on for the last two weeks – is just the probability that the third card drawn is the only one out of the remaining 79 that can be ‘joined’ with the first two, whatever they may be, i.e. 1/79 .

Since the game starts with 12 cards randomly placed on the table, there will be as many starts as there are combinations of 81 items taken 12 by 12, that is 81.80.79… 70/12!: about 70.7 billion.

The maximum number of cards that can be placed on the table without forming any combination is 20; Although the probability of this happening is already 19 less than one in a billion. As a curious anecdote, this is illustrated by Italian mathematician Giuseppe Pellegrino in 1971before creating the group (but that’s another article).

Regarding the average number of combinations we can expect with 12 cards on the table, this is what “featured user” Francisco Montesinos comments: “If we have 12 cards drawn at random, to form a combination, the first one of 12 methods can be chosen, the second of 11 and the third has a probability p = 10/79 of being in the remaining ten, so the possible number of combinations will be 12.11.(10/79)/6 = 220/79 = 2, 78″.

In the case of a mini deck, with only 3 properties and 27 cards, the probability that 3 randomly drawn cards will form a group will be 1/25, because, for any initial pair, only one of the remaining 25 cards will form a group with two of them.

### No wax over burns

With the Easter candles still burning, it seems like a good time to bring up some issues regarding candles and their burning times:

1. I lit a candle on Monday 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, every day I lit another candle and kept them burning for an hour… until today, when I ran out of candles. Knowing that it takes 4 hours for each candle to burn completely, what day is it?

2. “How fast time goes by! An old man says wistfully as he puts the candles away from my 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 your birthday?

3. We have two cylinder candles of the same height and the same material, but one is slightly thicker than the other, so that one takes 4 hours to burn completely and the other 5 hours. At some point, we extinguish both at the same time and notice that the height of the thickest candle is four times greater than the height of the other. how long has it been?

And finally, one very similar to the one taken from the XX Mathematical Olympiad:

4. We put two candles of different heights in the garden. The longest ones are 28 cm and take 7 hours to burn, and the shorter ones, which are thicker, take 11 hours to burn. We turn them on at the same time the party starts and 3 hours later, when friends leave, we turn them off. At that moment they are both the same height. What is the height of the shortest sail originally?

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