Background for the problem
I urge you to get a deck of cards and astonish yourself by following these simple
instructions. First, remove from the deck a set of five pairs of "mated" cards. (A mate
to a card is a card with the same value and color, i.e., the mate of the queen of
hearts is the queen of diamonds.) Call the five cards ABCDE and their mates abcde.
Arrange them in the order ABCDEabcde and place the packet of 10 cards facedown
on a table. You can then cut this packet of cards as many times as you want. (Cutting
a packet means lifting some cards off the top, thereby separating the packet into two
parts, and then transposing the two parts as you put them back together, always
keeping the facedown orientation of all the cards.) Then split the packet in half
(i.e., cut it into two sets of five cards each) and place the halves facedown on a table,
side by side. Then turn the five-card pile on the right faceup.
You are now going to spell the words in the phrase LAST TWO CARDS MATCH
in the following way. Pick up either pile (you can let someone else choose which
one) and spell the letter L by moving the top card of the pile to the bottom, and
then putting the pile back on the table without turning it over. Likewise spell the
letter A of LAST by once again picking a pile at random, moving its top card to the
bottom, and putting the pile back on the table, keeping the faceup or facedown
orientation of the cards. The spellings of S and T are done in the same way. After the
first word is completed, we still have two piles, one faceup and the other facedown.
Now remove the top card from each pile and set them aside, side by side, at some
vacant spot on the table. One will be facedown and the other will be faceup.
Repeat the process with the remaining piles of four by spelling the word TWO,
picking a pile at random for each letter. After the word is spelled out, remove
the top card from each pile and set them aside as before, side by side, below the
previously removed pair of cards. Proceed by spelling CARDS and MATCH with the
remaining piles, again removing the top cards after completing each word and
setting them aside.
You will now have two columns of cards (made up of the four sets of
removed cards), plus the two cards that are the last two cards left in your starting
piles. Focus on those last two cards, and turn over the one that's facedown.
Surprise-it matches the other card!
But that's not all. The trick has a second, even greater climax. Turn over
the remaining four facedown cards. Each one will match the faceup card it
is paired with!
Now here is the Problem of the Month:
First, let me state the instructions for the trick above in another way:
(1) Remove from the deck a set of five pairs of "mated" cards.
(2) Call the five cards ABCDE and their mates abcde.
Arrange them in the order ABCDEabcde and place the packet of 10 cards facedown
on a table.
(3) You can then cut this packet of cards as many times as you want.
(4) Then split the packet in half and place the halves facedown on a table, side by side.
(5) Then turn the five-card pile on the right faceup.
(6) Pick up either pile and move the top card of the pile to the bottom,
and then put the pile back on the table without turning it over.
Once again pick a pile at random, move its top card to the bottom,
and put the pile back on the table, keeping the faceup or facedown orientation of the cards.
Do this two more times. We still have two piles, one faceup and the other facedown.
Now remove the top card from each pile and set them aside, side by side, at some spot on the table.
One will be facedown and the other will be faceup.
(7) Repeat the process with the remaining piles of four but only move the top card to the bottom three times
(instead of four times as you did in Step (6). When finished, remove the top card from each pile
and set them aside as before, side by side, below the previously removed pair of cards.
(8) Repeat the process with the remaining piles of three but move the top card (from either pile) five times.
Then remove the top cards after completing the task and then set them aside.
(9) Repeat the process with the remaining piles of two again moving the top card five times.
The remove the top cards after completing the task and then set them aside.
(10) You will now have two columns of cards (made up of the four sets of removed cards),
plus the two cards that are the last two cards left in your starting piles.
Focus on those last two cards, and turn over the one that's facedown.
Then turn over the remaining four facedown cards.
Now, If you start with seven pairs of "mated" cards (instead of five), give the instructions for the trick.
You need only give the instructions for the number of times that you must remove the top card and place it on the bottom of the pile.
In the example for five mated cards, you would need only to send in the following:
When there are five cards in each pile, remove the top card from one of the piles FOUR times.
When there are four cards in each pile, remove the top card from one of the piles THREE times.
When there are three cards in each pile, remove the top card from one of the piles FIVE times.
When there are two cards in each pile, remove the top card from one of the piles FIVE times.
You get extra credit for coming up with a six word phrase to use while doing the trick!
Solution to the Problem:
The table below shows the number of times you must move the top card:You are now going to spell the words in the phrase LAST TWO CARDS MATCH in the following way. Pick up either pile (you can let someone else choose which one) and spell the letter L by moving the top card of the pile to the bottom, and then putting the pile back on the table without turning it over. Likewise spell the letter A of LAST by once again picking a pile at random, moving its top card to the bottom, and putting the pile back on the table, keeping the faceup or facedown orientation of the cards. The spellings of S and T are done in the same way. After the first word is completed, we still have two piles, one faceup and the other facedown. Now remove the top card from each pile and set them aside, side by side, at some vacant spot on the table. One will be facedown and the other will be faceup.
Repeat the process with the remaining piles of four by spelling the word TWO, picking a pile at random for each letter. After the word is spelled out, remove the top card from each pile and set them aside as before, side by side, below the previously removed pair of cards. Proceed by spelling CARDS and MATCH with the remaining piles, again removing the top cards after completing each word and setting them aside.
You will now have two columns of cards (made up of the four sets of removed cards), plus the two cards that are the last two cards left in your starting piles. Focus on those last two cards, and turn over the one that's facedown.
Surprise-it matches the other card!
But that's not all. The trick has a second, even greater climax. Turn over the remaining four facedown cards. Each one will match the faceup card it is paired with!
(1) Remove from the deck a set of five pairs of "mated" cards.
(2) Call the five cards ABCDE and their mates abcde.
Arrange them in the order ABCDEabcde and place the packet of 10 cards facedown on a table.
(3) You can then cut this packet of cards as many times as you want.
(4) Then split the packet in half and place the halves facedown on a table, side by side.
(5) Then turn the five-card pile on the right faceup.
(6) Pick up either pile and move the top card of the pile to the bottom,
and then put the pile back on the table without turning it over.
Once again pick a pile at random, move its top card to the bottom,
and put the pile back on the table, keeping the faceup or facedown orientation of the cards.
Do this two more times. We still have two piles, one faceup and the other facedown.
Now remove the top card from each pile and set them aside, side by side, at some spot on the table.
One will be facedown and the other will be faceup.
(7) Repeat the process with the remaining piles of four but only move the top card to the bottom three times
(instead of four times as you did in Step (6). When finished, remove the top card from each pile
and set them aside as before, side by side, below the previously removed pair of cards.
(8) Repeat the process with the remaining piles of three but move the top card (from either pile) five times.
Then remove the top cards after completing the task and then set them aside.
(9) Repeat the process with the remaining piles of two again moving the top card five times.
The remove the top cards after completing the task and then set them aside.
(10) You will now have two columns of cards (made up of the four sets of removed cards),
plus the two cards that are the last two cards left in your starting piles.
Focus on those last two cards, and turn over the one that's facedown.
Then turn over the remaining four facedown cards.
Now, If you start with seven pairs of "mated" cards (instead of five), give the instructions for the trick.
You need only give the instructions for the number of times that you must remove the top card and place it on the bottom of the pile.
In the example for five mated cards, you would need only to send in the following:
When there are five cards in each pile, remove the top card from one of the piles FOUR times.
When there are four cards in each pile, remove the top card from one of the piles THREE times.
When there are three cards in each pile, remove the top card from one of the piles FIVE times.
When there are two cards in each pile, remove the top card from one of the piles FIVE times.
You get extra credit for coming up with a six word phrase to use while doing the trick!
| Cards in the Pile | Number of Times to Move Top Card | |
|---|---|---|
| 7 | 6 or 13 or 20 | |
| 6 | 5 or 11 or 17 | |
| 5 | 4 or 9 or 14 | |
| 4 | 3 or 7 or 11 | |
| 3 | 2 or 5 or 8 | |
| 2 | 1 or 3 or 5 |
Of course, the easiest way to do this trick is just to remember this:
If there are N cards in each pile, just remove the top card and put it on the bottom N - 1 times. But the trick is better if you use the MOD function and choose a different number when you repeat the process. In the example, you could have done 4, 3, 2, and 1 instead of 4, 3, 5, and 5 (LAST, TWO, CARDS, MATCH) but using words or varying the number of times adds to the illusion.
If you REALLY want to impress your FRIENDS, you could do the trick with the FULL DECK. I must thank my wife for indulging me by doing the entire deck!! She liked the trick but she said it was too much counting! Here is how you set up the trick for the full deck:
First, arrange the 52 cards by suits from A, 2, 3, ... to J, Q, K. Then alternate red and black suits, combining the cards into one pile. Then proceed as before.
Have someone cut the deck as many times as they want to.
Then take the top 26 cards and place them face up, leaving the other 26 cards face down.
Then have the person remove the top card from either pile and put it on the bottom.
Have the person do it 25 times, then remove the top card from each pile and set them aside,
side by side, at some vacant spot on the table.
Now, repeat this process twenty-four more times. When there are 25 cards in each pile, remove the top card (from either pile) 24 times. When there are 24 cards, do it 23 times; when there are 22 cards, do it 21 times; all the way until there are just 2 cards in each pile, just do it once. Here is a partial table, showing how many times to remove the top card when you have a certain number of cards in each pile:
| Cards in the Pile | Number of Times to Move Top Card | |
|---|---|---|
| 26 | 25 | |
| 25 | 24 | |
| 24 | 23 | |
| 23 | 22 | |
| 22 | 21 | |
| 21 | 20 | |
| ...... | ...... | |
| 8 | 7 or 15 or 23 | |
| 7 | 6 or 13 or 20 | |
| 6 | 5 or 11 or 17 | |
| 5 | 4 or 9 or 14 | |
| 4 | 3 or 7 or 11 | |
| 3 | 2 or 5 or 8 | |
| 2 | 1 or 3 or 5 |
Correctly solved by:
| 1. Seth Parkinson |
Mountain View High School, Mountain View, Wyoming |