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Mathematics Behind “21 Card Trick”

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I have decided to center my mathematical exploration around the topic of Card tricks. Card tricks is one of the topic that has mathematical connections. First of all, I am going to explore the mathematics behind “21 card tricks”. After that, I will use probability and statistics to show the correlation between age/ gender and timing of people noticing the trick of “21 card trick” which is based on mathematics.

The major reason why I pick this topic is that Card tricks is one of my hobby since I was 12 years old. It all started when my friend’s father showed me the card trick “21 card trick”. It was quite impressive when I saw it for the first time and I started learning the trick from him. Consequently, I thought this is a good opportunity to investigate the mathematics behind the card trick.

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Before exploring the mathematics behind card tricks, it is important to understand the basic concept of playing card. Playing card consists of 54 cards in each of the four suits: Spades, Clubs, Diamonds and Hearts and 2 jokers. Each of the four suits contains Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen and King. Thus, we can calculate the total amount of card in one deck is cards I have decided to perform a card trick called “21 card trick”. This card trick is self-working trick which is based on mathematics. It does not require secret set up, trickery and other hidden move or magical move. Mathematics behind 21 card trick (self-working trick)

Procedure of “21 card tricks”

  1. Look for a volunteer or volunteers who can participate to the card trick.
  2. Make sure the jokers are removed and use only 21 cards out of 52 cards.
  3. Shuffle all the 21 cards in any order. You can ask one of the volunteer to shuffle so that they can cut the deck as much as desired.
  4. After shuffling the cards, ask the volunteer to pick only one card without showing you which card they have chosen.
  5. Shuffle the cards again.
  6. Divide the 21 cards into three columns. Make sure you deal the cards row by row and starting from left to the right. E.g. 1st card into the first pile, 2nd card into the second pile, 3rd card into the 3rd pile and then 4th pile into the 1st pile. Each of columns have 7 cards. Make sure they are all face up so we can see the cards.
  7. Ask the volunteer to point out that which of the columns contains the selected card.
  8. Gather up the three piles into one piles again. In this time, you need to make sure to put the designated column into the middle of the three columns and do not shuffle.
  9. Repeat steps 6, 7 and 8.10. Again repeat 6 and 7.
  10. The middle card in the pointed column is the card that the volunteer selected in Step 4.

Mathematical trick behind “21 card trick”21 cards are placed in 7rows 3columns.whenis the position of the selected card when the column is in the middle. The selected card in the row is placed in the middle of the column when collected.

Probability behind the trick

In this trick, every time the volunteer point out the column which contains the selected card, it narrows down the options. This can be explained by using probability.

  1. In the first step, there are 21 cards so that the probability of getting the selected card is.
  2. After you are first told a column which consist of 7 cards, you can narrow it down to. Now probability is.
  3. Then next step you can narrow it toor
  4. After you are third time told the column, you can find the selected card which gives probability of

Statistics

I did the 21 card tricks to 31 people (15 females and 16 males) including my family, friends and acquaintances. In order to estimate the timing of people noticing that the trick is self-working, I will ask them a question “Did you notice any pattern?”. If the person says “yes” and explain that the trick is behind mathematics, then I will estimate that it is the timing they notice the trick.

Gender

I have found that there is no significant difference between female’s timing of noticing the trick and male’s timing of noticing the trick. According to Figure 1 and 2, the number of male who notice the trick when they see it for first time is 6 and female is 3. This data shows that male has more ability find the trick than female has. The scatter plot above shows that age increases the timing of people noticing the trick of “21 card trick”. This means that older people need more time to find the trick and younger people need less time to find the trick. As shown in Figure 3, the regression line shows that the relationship between the age of the volunteers and their timing of noticing the trick of “21 card trick” is in positive direction. In addition, the strength of the scatter plot is moderately strong and there is no remarkable outlier in the data. I have used the system of Excel to draw the regression line which is the line of best fit. As a consequence, we can find that there is a correlation between the age of volunteers and the timing they notice the trick. The equation that can be used to approximate timing of people noticing the trick when given their age is y = 0.0863x – 0.1768. e.g.

Let’s say person who are 20 years old is chosen for the volunteer of the card trick. Then x=20. Approximately 1.5 time is needed for finding out the trick. Let’s say person who are 85 years old is chosen for the volunteer. Then x=85Approximately 7.2 times is needed for finding out the trick. In conclusion, I have found that the relationship between age and timing of noticing the trick is positive. I could have done better on data gathering because I only had 31 peoples’ data which is not enough and not accurate to prove that there is correlation between age and the timing.

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