4. To practice his counting, my grandson invented a card game he calls “Facetime”. He takes two decksof 54 cards (including the jokers) and lays them all out on the floor in a line, face up, mentally numberingthem (1, 2, 3, etc., up to number 108). He then turns every other card face down, all the way down theline (so then, all the even-numbered cards are face down). Next he looks at every third card and flips it:if it’s face down, he turns it face up, and vice versa, all the way down the line. He then flips every fourthcard, and then every fifth. He continues this way, counting by every number (by 6’s, then by 7’s, etc.)until he has finished counting by 108’s (which doesn’t take long, since that’s how many cards there are.)When he’s done, which cards are lying face up?

Question
Asked Dec 3, 2019
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4. To practice his counting, my grandson invented a card game he calls “Facetime”. He takes two decks
of 54 cards (including the jokers) and lays them all out on the floor in a line, face up, mentally numbering
them (1, 2, 3, etc., up to number 108). He then turns every other card face down, all the way down the
line (so then, all the even-numbered cards are face down). Next he looks at every third card and flips it:
if it’s face down, he turns it face up, and vice versa, all the way down the line. He then flips every fourth
card, and then every fifth. He continues this way, counting by every number (by 6’s, then by 7’s, etc.)
until he has finished counting by 108’s (which doesn’t take long, since that’s how many cards there are.)
When he’s done, which cards are lying face up?

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Expert Answer

Step 1

Two decks of 54 cards (including the jokers) are taken. He lays them all out on the floor in a line, face up, mentally numbering.

In one deck, there are 52 cards + 2 jokers = 54 numbers of cards. There are two decks which means 54+54=108 number of cards.

Step 2

 He then turns every other card face down, all the way down the line (so then, all the even-numbered cards are face down). Next he looks at every third card and flips it:
if it’s faces down, he turns it face up, and vice versa, all the way down the line. He then flips every fourth card, and then every fifth. He continues this way, counting by every number (by 6’s, then by 7’s, etc.) until he has finished counting by 108’s (which doesn’t take long, since that’s how many cards there are.).

So, to find out the faces after the above procedure, we will follow the following trick.

  • 1, 2, 3, … , 108 are Face Up.
  • 1 is divisible by 1 only. So, the face will change 0 times during the process. Thus, the final face for 1st card is Face Up.
  • 2 is divisible by 2. So, the face will change 1 time during the process. Thus, the final face for 2nd card is Face Down.
  • 3 is divisible by 3. So, the face will change 1 time during the process. Thus, the final face for 3rd card is Face Down.
  • 4 is divisible by 2 and 4. So, the face will change 2 times during the process. Thus, the final face for 4th card is Face Up.
  • 5 is divisible by 5. So, the face will change 1 time during the process. Thus, the final face for 5th card is Face Down.
  • 6 is divisible by 2, 3 and 6. So, the face will change 3 times during the process. Thus, the final face for 6th card is Face Down.
  • 7 is divisible by 7. So, the face will change 1 time during the process. Thus, the final face for 7th card is Face Down.
  • 8 is divisible by 2, 4 and 8. So, the face will change 3 times during the process. Thus, the final face for 8th card is Face Down.
  • 9 is divisible by 3 and 9. So, the face will change 2 times during the process. Thus, the final face for 9th card is Face Up.
  • 10 is divisible by 2, 5 and 10. So, the face will change 3 times during the process. Thus, the final face for 9th card is Face Down.
  • And so on.
Step 3

It can be noticed from the above step that the face is up in the places where the number is a square number. So, the cards 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 are lying face up when there are 108 cards in total.

When th...

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