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(Message started by: Mugwump101 on May 10th, 2004, 11:55am)
Title: Mathemathics Magic Trick
Post by Mugwump101 on May 10th, 2004, 11:55am
Hi, sorry to be a bother again -_-' But my teacher demenstrated-sp?- a Math Magic trick with us today. She said Magic Tricks usually involve a lot of Math, and we should guess what type of Math and patterns to find in her trick. I keep thinking about it but it doesn't work out, and we can't memorize the #s so I can't list all of them, but I think I know two charts of them.
Okay, here is how it goes. She said to pick a positive integer less than 32. When a student picks a # she says show it to the class using his/her fingers. And She turns aways covering her eyes with the charts in the other direction.
When the students shows let's say 12. After s/he is done showing the number, the teacher shows the class I think five charts. Each chart shows several numbers. The #'s can repeat on some of the charts.
I memorized (some #s but added some to follow a pattern) only two of the five charts.

Chart One:                    There was also another chart but
4 5 6 7                          I only know the beginning:
12 13 14 15                                          
20 21 22 23                                        8 9 10 11
28 29 30 31                                        12 13 14 15

Chart Two:
2 3 5 7
10 11 13 15
18 19 21 23
26 27 29 31

Something along those lines.... If you remember any pattern of this sort. Can you please explain to me? Thank you so much for your time.

NOTE: I don't know if all the #s were correctly arranged like this because I just followed the pattern of downwards of +8 and +4 (just for the beginning one). Some #s could be misleading or wrong.


Title: Re: Mathemathics Magic Trick
Post by towr on May 10th, 2004, 12:27pm
What does she do with the chart after she shows it to the class? Ask for each if the number is on it?
if so 5 charts = 5 bits = 2^5 = 32 numbers you can distinguish
Simple binary counting would work
1 = 00001 (no,no,no,no,yes)
2 = 00010
3 = 00011
4 = 00100
etc
Title: Re: Mathemathics Magic Trick
Post by Sir Col on May 10th, 2004, 1:17pm
Was it like this (http://mathschallenge.net/index.php?section=rec&ref=general/experiment_4&type=general)?

If so, I can give you some hints as to how it works; although towr has already given a big hint.

If you want to see some other "mind reading" (http://mathschallenge.net/index.php?section=rec&ref=about&type=general) puzzles in action, I have a few others on my website.
(the "grid" trick is experiment 4)
Title: Re: Mathemathics Magic Trick
Post by Mugwump101 on May 10th, 2004, 2:00pm
Um.... She doesn't ask for each if the # is on it. She just asks the person who voleentered to come up and pick a positive integer less than 32 (She does this so the class would know the # the person has in mind incase s/he says the teacher didn't guess the # although she did but s/he changed it in his mind and no one would know....etc... I hope you understand what I just explanded sorry if you didn't)
  5 bits? Why is 2^5? sorry to not understand it. I'll check out the website too Sir Col.
Title: Re: Mathemathics Magic Trick
Post by towr on May 10th, 2004, 2:28pm
Ok, let's go through this..
-She has 5 charts
-She ask someone to pick a number (which she doesn't see/hear)
-Then obviously she must do something with those charts, as they're not there for decoration and she's not psychic
-She reveals the number (correctly) and everyone gazes in amazement ;)

An obvious way to design the charts would be to make one with the numbers 1-16
one with 1-8 and 17-24
one with 1-4, 9-12, 17-20 and 25-48
one with 1-2, 5-6, 9-10, 13-14, 17-18, 21-22, 25-26, 29-30
and a 5th one with only the odd numbers

This way when you're told on what charts the chosen number occurs you can easily find which number it must have been. This is because each chart cuts the number of possible numbers it might be in 2.

Suppose the number to be found is 12
Before we start, as far as the teacher knows it may be any number from 1 to 32.
Now if she is told the number occurs on the first chart, then she knows it must be from the range 1-16
If it isn't on the second chart then it must also be from either in the range 9-16 or the range 25-32, so that leaves 9-16
It's on the third chart, so it must be in 9-12
It's not on the fourth chart so it can only be 11 or 12
And as it's not on the fifth chart it's not odd and so can only be 12

The charts your teacher uses probably work on the same principle, but the numbers are distributed in a different (less obviously binary) way over the 5 charts.
Title: Re: Mathemathics Magic Trick
Post by Mugwump101 on May 11th, 2004, 8:24pm

on 05/10/04 at 14:28:54, towr wrote:
Ok, let's go through this..
-She has 5 charts
-She ask someone to pick a number (which she doesn't see/hear)
-Then obviously she must do something with those charts, as they're not there for decoration and she's not psychic
-She reveals the number (correctly) and everyone gazes in amazement ;)

An obvious way to design the charts would be to make one with the numbers 1-16
one with 1-8 and 17-24
one with 1-4, 9-12, 17-20 and 25-48
one with 1-2, 5-6, 9-10, 13-14, 17-18, 21-22, 25-26, 29-30
and a 5th one with only the odd numbers

This way when you're told on what charts the chosen number occurs you can easily find which number it must have been. This is because each chart cuts the number of possible numbers it might be in 2.

Suppose the number to be found is 12
Before we start, as far as the teacher knows it may be any number from 1 to 32.
Now if she is told the number occurs on the first chart, then she knows it must be from the range 1-16
If it isn't on the second chart then it must also be from either in the range 9-16 or the range 25-32, so that leaves 9-16
It's on the third chart, so it must be in 9-12
It's not on the fourth chart so it can only be 11 or 12
And as it's not on the fifth chart it's not odd and so can only be 12

The charts your teacher uses probably work on the same principle, but the numbers are distributed in a different (less obviously binary) way over the 5 charts.


Whoa, that's amazing! This is the best explanation to me! Thank you so much Towr! This was a Very interesting conclusion and theory ^_^ Thank you!
Title: Re: Mathemathics Magic Trick
Post by rmsgrey on May 12th, 2004, 2:17am
The advantage of using binary charts for this trick (ones where if you bitwise or the binary representations of the numbers on a given chart together, you get a number with precisely one bit set - a power of two if you prefer) is that it's easy to calculate the number just by knowing which charts it's on - the lowest number on each chart is a power of two, and summing those powers of two that correspond to the appropriate charts gives you the number.

The disadvantage of binary charts is that anyone who knows binary numbers is liable to catch on pretty quickly.

The advantage of using a random number generator to generate the charts (pick half the numbers in your range at random, put them on one chart. Pick half the numbers on that chart, and half the numbers not on that chart, and put them on a second chart. Pick half the numbers on both charts, half those on the first but not the second, half those on the second but not the first and half those on neither to make the third chart. Continue doing this until you get down to choosing between two numbers in each set for the last chart) is that it's much harder to spot the trick.

The disadvantage of randomly generated charts is it takes much more work to identify the number from a list of charts - since most of the impressiveness of the trick comes from the speed with which the number is identified, the answer is usually to compromise - pick a slightly less obvious pattern than straight binary, but one that's still easy to calculate.
Title: Re: Mathemathics Magic Trick
Post by Leon on May 12th, 2004, 10:55am
There is a card trick that does the same thing. I don't recall it exactly, but it was something like:

4 rows of 4 cards (which cards doesn't matter). The particpant picks a card and identifies which column it is in. The delaer then picks up a non-selected column, followed by the selected column, then the other two. The dealer than deals it back out and has the partcipant pick the row. Dealer picks them up in same fashion. After doing this 3 or 4 times, the selected card is the 7 card (or so), which can be done in dramatic fashion. Oohs and Aaahs all around.
Title: Re: Mathemathics Magic Trick
Post by rmsgrey on May 13th, 2004, 4:33am
I know a version of that one with 27 cards - deal into 3 piles, find out which pile the selected card is in, pick the piles up and deal again. After finding out which pile the card's in, pick them up again and deal them a third time. After finding out which pile the card is in, pick them up again, face down, and fan them out briefly, then reveal the chosen card.

By varying the order in which the piles are picked up, it's possible to put the card into any chosen position (ternary coding)
Title: Re: Mathemathics Magic Trick
Post by Mugwump101 on May 13th, 2004, 12:56pm
      wow, it's cool how there are different variations to game. Leon and rmsgrey can you go into more detail like showing me an example and how to do it from there? I want to try it at home, and maybe work out a game with my friends ^_^ It'll be fun.
      I love your explanation of advantages and disadvantages of Binary Numbers rmsgrey.
     btw, I told my teacher today about the charts, and it uses the binary concepts, and she's like How you know that? I told it's a secret, but she cajoled me, and I told her about a Riddles Forum of Mathemathics, Riddles, etc.. Then she asked me how would you notice the binary #s without someone telling you like to notice it in class.
  Well, I didn't know how to response to that as a whole explanation but I have a general idea and I want to know if that's true.
 
    I wanted to tell her something along the lines of I can spot the even #s on several of the charts as halves of the number before. How I go futher into explanation for that? ( I know how she does it because Towr wrote a great explanation on how. Should I use that in my explanation for finding it easly in the classroom? Thank you again!!!)
      And lastlySir Col, everytime I open your website, the website freezes on my computer (I'm thinking along the lines of it has too much graphics or something. Do you know a solution as to how I can fix that?)


Title: Re: Mathemathics Magic Trick
Post by Leon on May 13th, 2004, 1:54pm
I figured out how the trick works, didn't even take long to remember it.

Deal 15 cards in 3 columns going left to right and top to bottom:

1    2    3
4    5    6
7    8    9
10  11  12
13  14  15

Ask the person to pick a card, but not tell you. They only need indicate which column it is in. Column 1 (left) 2 (middle) or 3 (right).

Pick up the columns on by one with their selected column second. Let's say the card is #10, so they said column 1. Stack column 3 in order face down (3,6,9,12,15) followed by column 1 and then column 2. So face down you have the 15 cards in the following order: 3,6,9,12,15,1,4,7,10,13,2,5,8,11,14.

Now deal the cards out face up in the same fashion (top left across and then down to a new row). The cards would look like:

3    6    9
12  15   1
4    7   10
13   2   5
8    11 14

Ask the person what column it is. Since their card is #10, they will say column 3. Pick up column 2, then 3, then 1 (not the only thing that matters is column 3 is 2nd) and deal them out again:

6   15   7
2   11   9
1   10   5
14  3   12
4    13  8

Do it one more time (pick up columns 1,2,3 in order) and you end up with:

6    2    1
14   4   15
11  10   3
13   7   9
5    12   8

The selected card will be in the middle everytime (it does appear there sometimes after round 3, but not always (for example try it with #15 and it takes until rd 4) I'm not sure why,  I haven't look at it to lcosely). You can reveal the card in any fashion. Stack them and Count to #3, #8 or #13 depending on how you stacked it or fan and pcik it out yourself, etc.

Unless I've screwed something up in the ordering, that is it.
Title: Re: Mathemathics Magic Trick
Post by Sir Col on May 13th, 2004, 2:25pm

on 05/13/04 at 12:56:04, Mugwump101 wrote:
And lastly Sir Col, everytime I open your website, the website freezes on my computer (I'm thinking along the lines of it has too much graphics or something. Do you know a solution as to how I can fix that?)

Really? It is certainly not graphic hungry; in fact, it is quite the opposite. I am often the butt of my friends' jokes who complain how "boring" the presentation is!

Is anyone else experiencing a similar problem: computer freezing, or know why MugWump101 might be having this problem?

The link for the "mind reading" experiments is:
http://mathschallenge.net/index.php?section=rec&ref=about&type=general
Title: Re: Mathemathics Magic Trick
Post by rmsgrey on May 17th, 2004, 4:05am
The first time I came across the number guessing trick used the binary version, which was pretty obvious.

For the redealing cards, if you use 27 cards, and always pick up the pile with their card second, then, at the start, their card will be in positions 1-27. After picking up the piles, their card will be in positions 10-18, which will get dealt out as the middle 3 in each pile, so ends up in positions 13-15 after picking up the second deal. On the third deal, positions 13-15 get dealt out as the fifth card in each pile, so when you pick up the piles, you put their card at position 14.

If you use ternary coding, then you can easily pick which position you want their card to end up with. The positions are labeled from 000 (0) to 222 (26) according to the number of cards above them. Work out the code for the position you want, so, if you want to put the card to be the seventh card in the pile, that's six cards above it. Six is 020 in ternary, so the code you want is 020. After each deal, pick up the piles according to the code digit so that their card ends up in the top pile (when the pack is turned face down) for a zero, in the middle for a one, or on the bottom for a two. Read the digits from right to left to work out which order the deals come in - the first deal is picked up according to the last digit of the code, etc. For instance, if the code is 102 (11) then the first deal gets picked up so that their card's pile is on the bottom (2), the second deal puts their card's pile on top (0), and the third deal picks up to place their card's pile in the middle (1). Throw away the top 11 cards, and the next will be their card.
Title: Re: Mathemathics Magic Trick
Post by Mugwump101 on May 20th, 2004, 6:27pm
wow, interesting. Thanks for your tips. Tenary coding sounds pretty cool though it's rather hard for me to get.

I was wondering, I had assigned Math Homework today that says create a Math Game. I wanted to choose Leon's game but she said write the idea and objective of the game but it has to be in the topics we're learning in 8th grade. I'm rather baffled as to what topic it belongs too -_-' Can you be a help?

btw, Towr, I told my teacher and showed her your way but she said you're on the right track, it is binary and etc... but it uses simplier terms than that. Can you tell me what way can you find it in the classroom using simplier methods? Thanks a lot for your help.
Title: Re: Mathemathics Magic Trick
Post by towr on May 21st, 2004, 12:57am

on 05/20/04 at 18:27:26, Mugwump101 wrote:
wow, interesting. Thanks for your tips. Tenary coding sounds pretty cool though it's rather hard for me to get.
You can use other bases as well. Perhaps base 5 might be easier, as it's more like 'normal' (base-10) numbers.
And of course you can adapt the same card game to use it, by using 5 coloumn instead of three.


Quote:
I was wondering, I had assigned Math Homework today that says create a Math Game. I wanted to choose Leon's game but she said write the idea and objective of the game but it has to be in the topics we're learning in 8th grade. I'm rather baffled as to what topic it belongs too -_-' Can you be a help?
Not without knowing which topics you have in 8th grade.


Quote:
btw, Towr, I told my teacher and showed her your way but she said you're on the right track, it is binary and etc... but it uses simplier terms than that. Can you tell me what way can you find it in the classroom using simplier methods? Thanks a lot for your help.
I really can't say without knowing all the charts she used. Or perhaps what topic it's related to.
You could make the charts so that you just have to add the first number of each chart you target is on, like rmsgrey said earlier. But I don't see how that makes the terms involved in constructing the charts any simpler (only the use of them)
Title: Re: Mathemathics Magic Trick
Post by Sir Col on May 21st, 2004, 1:18pm
I imagine your teacher uses charts similar or identical to the charts on my website.

The method is very simple. To demonstrate the idea I shall show how you can work the trick with picking a number from 1-16.

First of all the hard stuff (THEORY)
Begin by writing 0-15 in binary:
0000=0
0001=1
0010=2
0011=3
0100=4
0101=5
0110=6
0111=7
1000=8
1001=9
1010=10
1011=11
1100=12
1101=13
1110=14
1111=15

As there are 4 bits (0's and 1's), it will require four charts.

Chart 1 contains all the numbers for which the 4th bit is 1:
1,3,5,7,9,11,13,15

Chart 2 contains all the numbers for which the 3rd bit is 1:
2,3,6,7,10,11,14,15

Chart 3 (2nd bit = 1):
4,5,6,7,12,13,14,15

Chart 4 (1st bit = 1):
8,9,10,11,12,13,14,15


Now the easy stuff (METHOD)
After picking your number, the teacher asks if your number is in each chart. If your answer is, "Yes," (s)he will add the first number in each chart to a running total; the first numbers in each chart being 1, 2, 4, and 8. The final total is the number you picked.

To disguise the method further (like I did on my website), and make it work for the numbers 1-16 (rather than 0-15), one is added to every number in the chart. The numbers in the top left would now be 2, 3, 5, 17. Once the teacher has the total, they subtract one to obtain your number. It also helps to shuffle the order of the charts, as the only thing that is important is the first number on each chart.

The other option, which your teacher seems to have done, is not add one and allow you to pick a number from 1 to 32 (rather than 0 to 31). If you say, "No," to all the charts then your number is 32.

I prefer my method as it is slightly more subtle and hides the link with binary.


To make charts the same as your teacher, you would need to use five charts and write out the binary representations of the numbers 0-31.


The five charts your teacher probably uses would be as follows:

(1) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31
(2) 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31
(3) 4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31
(4) 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31
(5) 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31

Try it out! Pick a number from 1 to 32, then look if it is in each list. If it is, add the number at the beginning of each list to your total. The final total is your number.

For example, if you picked 11, you would find it in the 1st, 2nd, and 4th chart. Checking: 1+2+8=11.
(Remember if the number isn't in any chart then the number will be 32.)
Title: Re: Mathemathics Magic Trick
Post by Mugwump101 on May 23rd, 2004, 7:31pm

on 05/21/04 at 13:18:59, Sir Col wrote:
I imagine your teacher uses charts similar or identical to the charts on my website.

The method is very simple. To demonstrate the idea I shall show how you can work the trick with picking a number from 1-16.

First of all the hard stuff (THEORY)
Begin by writing 0-15 in binary:
0000=0
0001=1
0010=2
0011=3
0100=4
0101=5
0110=6
0111=7
1000=8
1001=9
1010=10
1011=11
1100=12
1101=13
1110=14
1111=15

As there are 4 bits (0's and 1's), it will require four charts.

Chart 1 contains all the numbers for which the 4th bit is 1:
1,3,5,7,9,11,13,15

Chart 2 contains all the numbers for which the 3rd bit is 1:
2,3,6,7,10,11,14,15

Chart 3 (2nd bit = 1):
4,5,6,7,12,13,14,15

Chart 4 (1st bit = 1):
8,9,10,11,12,13,14,15


Now the easy stuff (METHOD)
After picking your number, the teacher asks if your number is in each chart. If your answer is, "Yes," (s)he will add the first number in each chart to a running total; the first numbers in each chart being 1, 2, 4, and 8. The final total is the number you picked.

To disguise the method further (like I did on my website), and make it work for the numbers 1-16 (rather than 0-15), one is added to every number in the chart. The numbers in the top left would now be 2, 3, 5, 17. Once the teacher has the total, they subtract one to obtain your number. It also helps to shuffle the order of the charts, as the only thing that is important is the first number on each chart.

The other option, which your teacher seems to have done, is not add one and allow you to pick a number from 1 to 32 (rather than 0 to 31). If you say, "No," to all the charts then your number is 32.

I prefer my method as it is slightly more subtle and hides the link with binary.


To make charts the same as your teacher, you would need to use five charts and write out the binary representations of the numbers 0-31.


The five charts your teacher probably uses would be as follows:

(1) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31
(2) 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31
(3) 4, 5, 6, 7, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, 31
(4) 8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31
(5) 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31

Try it out! Pick a number from 1 to 32, then look if it is in each list. If it is, add the number at the beginning of each list to your total. The final total is your number.

For example, if you picked 11, you would find it in the 1st, 2nd, and 4th chart. Checking: 1+2+8=11.
(Remember if the number isn't in any chart then the number will be 32.)


This was the exact idea I was looking for ^_^ THANKS A BUNCH!! I'm pretty sure this has to be the easier way she's always talking about ^_^ YAY! I tried it at home and it works great! THANK YOU!

btw, Towr the topics are like um... Probability, measurement, algebric equations etc... like it's the pre-Math A stuff. I just wanted to know what the topic of the game would go under as such as in this case.
      It uses some kind of 4-step method to obtain the # column card in the middle. Actually, I very curious as to how that works. How does the card in any column while placing it in the middle work out as the middle card in 4 turns? Does that limit ways using the binary system. If so, how so?

Thank you SO SO SO much for helping me!! THANK YOU!!


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