**DEADLINE: SUNDAY, APRIL 10TH, 2011, 23:59:59**

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# 1 A Magic Effect with a Prepared Deck

On Friday 24th, March, 2011, a magical effect was performed in class. This is what happened. The professor produced a deck from his pocket. He asked for two volunteers, Luis and Jara and asked one of them to riffle shuffle the deck. Before actually proceeding to rifle shuffle the deck, he clarified that we would do perfect riffle shuffle. This kind of riffle shuffle implies that the deck is divided into two packs of 26 cards each. Jara volunteered to do it and she certainly did it gracefully. After that, the professor dealt out 10 cards to each student and made the following predictions.

“Jara,” the professor said, “your two first cards have different colours.”

She giggled and replied, “That’s quite easy as you get half of the chances.”

“Then, I’ll make it more complicated. Luis, I foretell you also have two cards with different colours.”

They checked their cards and found he was right.

“I’ll tell you more. The next two cards of each of you are also of different colours,” added the professor merrily.

More incredulously, they turned the two next cards and checked they were of different colours. They nodded approval.

“Now, take four cards each of you.”, continued the professor. “What are the odds both get the four suits?”

“No way!,” they both exclaimed their surprise at the same time.

Once more, they turned their cards and found, to their amazement, that they had the four suits. They looked at each other jollily.

“Even more,” added the professor, “if you take two cards each, you’ll gather the four suits.”

“Surely enough!,” said Luis, not without a certain touch of contempt. If there was no sleight of hand, then this was going too far for Luis.

But that was the case: the four suits were in their hands. The professor checked the remaining 6 cards and announced their colours would alternate, and again he was right. Next, the professor counted 13 cards

and handed over to Luis and the remaining packet to Jara.

“Luis, please, remove the face cards.” He did so. “Now, sum all the value of the remaining cards.”

While Luis was summing those values, the professor took two sheets of papers; each one had a big “5” written on it. Luis finally announced that the total is 55. The audience laughed quite good-humouredly.

“Jara, by any chance, won’t you have an ace in your set of cards?,” asked mysteriously the professor. She searched slowly and found an ace.

“What about a two?” She searched and found a two. “How nice of you! What about a three?” She searched and found a three.

It turned out she had all the cards values from ace to king. Standing ovation.

# 2 Questions about the Magic Effect

The student is asked the following:

- Find out how the professor prepared the deck. Watch the video again to gain insight of the counting and how this was done.
- Describe the mathematical ideas behind the magic effect.
- Write a report detailing your ideas.

# 3 Written Paper

A paper describing the following points must be handed over.

- Description of the magic effect.
- Proof of your mathematical idea.
- The paper has to be written in correct Spanish or English; it also has to possess clarity of thought. Show me what you know; do not force to search for it through a poorly written paper.

# 4 Grading

This projects counts 0.25 of your final grade.

# 5 Questions and Office Hours

I am willing to answer your questions during my office hours.