Logic Application, Statistics Problem Example

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Statistics problem

Salient Facts

This problem is defined by important salient facts that make it possible to make the application of logic to obtain the solution of the problem. These salient facts are defined by the rules of the game and the value of the cards that each of the player possess and the question cards that any of the players draw at any given point during the game. The salient facts are what will guide any player to formulate a logical strategy and also employ basic mathematical statistical methods, which is probability, to obtain all the values of their three cards.

The different players of the game have the following cards:

Player Card 1 Card 2 Card 3
Andy 1 5 7
Belle 5 4 7
Carol 2 4 6
Me ? ? ?

 

  • Andy can only see Belle’s, Carol’s and my cards
  • Belle can only see Andy’s, Carol’s and my cards
  • Carol can only see Andy’s, Belle’s and my cards
  • I can only see Andy’s, Belle’s and Carol’s cards

Strategy

The only logical strategy to solve this problem would be:

Firstly derive the possible sum value of my cards from Andy’s observations. That means that since Andy can see Belle’s, Carol’s and my cards, and that he sees two players with the sum value of their cards, then my cards’ sum value is equal to either Belle’s or Carol’s sum value.

Finally, I can derive the value of my third card mathematically from Belle’s observations. Belle can see all the odd numbers, that is, she can see 1, 3, 5, 7 and 9. Since Belle cannot see her own cards, she can only see Andy’s Carols and my cards. Since Carol has the cards 2, 4 and 6 which are all not prime numbers then it means that all the prime numbers that Belle sees are between Andy and I. Here, I can apply mathematical techniques to find out the value of my third card.

Step-by-Step Solution

Step 1

If Andy can see 2 or more players whose cards sum up to the same value, then it means my cards’ sum total equals either Bell’s or Carol’s. The sum value of Belle’s cards is 16 while the sum value of Carol’s cards is 12. We keep in mind that Andy cannot see his cards.

Step 2

If Belle can see all of the 5 odd numbers, then it means that 1, 3, 5, 7 and 9 are visible to Belle. Considering that Andy has 1, 5 and 7, while Carol has 2, 4 and 6, (that is she has no odd numbers), then I have both 3 and 9 as two of my cards. This is keeping in mind that Belle cannot see her cards.

Step 3

Considering that Andy has already given me the possible sum of my numbers, I can mathematically calculate the value of my third card. Since the possible sum value of my numbers are either 12 or 16, I can make a logical conclusion. Adding the value of my two cards 3 and 9 I get 12, then the only logical sum value of my cards is 16. Subtracting the sum value of my two cards, 12, from 16, that is 16 – 12, I get 4. My third card is 4

Solution

The value of my three cards are 3, 4 and 9.

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