Monday Morning Math: Three Number Tricks

Hello everyone!  It’s April, so time for some tricks!  Technically it’s past April Fool’s Day, but these aren’t the prank kind of tricks anyway – just a few quick fun things for a happy start to the week.

Trick 1:  Quickly guessing the number! (from thoughtco.com)
Pick a 3-digit number where all the digits are the same (like 777).
Add those digits.  Divide your original number by this sum.  Did you get…..37?  You did!

Warning:  You can probably only do this trick once before people catch on.   Or you could go fancy and do a similar trick with a 9-digit number.  In that case instead of 37 you’ll get 12345679 which I admit both surprises and delights me.  

Trick 2: Elaborate guessing the number! (from our own blog)
Pick a 3-digit number, like 360.  For this the first and last digit have to be different.
Now reverse the digits (063), and subtract the smaller number from the larger (leaving 297).
Take that new number, reverse its digits, and add those two together (297 and 792).  I got 1089 when I did that but the cool thing is, so did you!  Even if you started with a different number!!!  

Trick 3: Guessing your age  (from our notes for one of our classes.)
Pick the number of days a week that you like to go out.  Now double this number.  Add 5, and then multiply by 50.  You’ll have a pretty big number now, but you’ll need to make it even bigger. 
If you have already had your birthday this calendar year, add 1773.  If you have not had a birthday this calendar year, add 1772. 
Finally, subtract the four-digit year that you were born.   
You’ll end up with a three-digit number: The first digit of this was your original number (i.e. how many times you want to go out a week).  The second two digits are your age!!!  Pretty neat huh?  The only catch with this is that this trick changes every year – you’ll have to add a diferent amount if you try it in 2024.

Happy trickstering!

More Math Mindreading

Godzilla is a well-known mind-reader, and in honor of final exams, which are coming up sooner than seems possible, he’d like to demonstrate his powers.  Even over the internet, because his powers are MIGHTY.  Like him.

Start with 3-digit number that is not a palindrome (so 360 is OK, but 363 is not).    Then reverse the digits, and subtract the smaller number from the larger.  You get a NEW AND IMPROVED number.  So if you do start with 360, your NEW AND IMPROVED number will be 297 (which is 360 minus 063).

Treating your NEW AND IMPROVED number as a 3-digit number, reverse the digits.   This means that if your NEW AND IMPROVED number appeared to only have two digits, or even one, then you have to tack on one or two leading zeros that you include in the reversal.

Now add your NEW AND IMPROVED number to its reverse.  Godzilla will now tell you the sum, even over all the miles and electrons that separate you from this friendly beast…..

Illusion Knitting

See Mini-G look at this fine piece of stripey art:

Isn’t that interesting, full of nuance?  NO — it looks totally boring.  But Mini-G is actually looking at it at an angle, which turns out to be a completely different story.

No more simple stripes!  And while it’s no Mona Lisa*, it’s pretty cool to see the shapes appear just as you start to walk away in search of something less vertical to look at.  Even better, it’s simple knitting.  REALLY simple knitting, just knits and purls, where using stockinette stitch makes a color fade into the background when viewed from the side, and using garter stitch makes a color stand out.  There’s a great explanation here, where “great”=“uses legos”.

This comes from Woolly Thoughts (“In pursuit of Crafty Mathematics”) and their newish illusion site.  It’s a free pattern — Woo hoo! — and not that I’m suggesting that you knit during meetings or anything, but if you DID knit during meetings this particular pattern is simple enough that you can do it without being distracted from the Important Conversations and Presentations, and then you can feel good at the end of two hours that you made quite a bit of progress on your knitting whether the meeting led to a resolution or not, plus you get to point out that you’re really doing mathematics if anyone asks what you’re knitting.  Win-win!

* though there is a pattern for that.

Math Magic: Predicting the final card

Here’s a slightly more impressive math card trick than yesterday. Start with a full deck of 52 cards [or, rather, start with any 52 cards — it’s not a problem if they come from a mixture of decks]. Give the deck to a Volunteer to shuffle, and afterwards have the Volunteer take the top 12 cards and give you the rest. Sometime during the next step you should sneak a peek at what the bottom card is in the rest-of-the deck. This can best be done by holding your hand casually so the bottom card is face-up enough for you to see, but it looks like you’re just holding them up before placing them down at the end of the next step.

The Volunteer should then look at their 12 cards, not showing them to you, and pick any four of them. [This is a good time to be glancing at the bottom card in the pile you hold in your hand!] Those four get laid face-up in four separate piles on the table, and the poor unchosen eight get put face down in another pile. You put the rest of the deck on top of that pile.

Now think a moment, perhaps idly shuffling the top bunch of cards in the pile [just not the bottom 9], and say that you are going to have the Volunteer lay out some cards and you will predict the last one. Concentrate, and then write the name of the bottom card that you so sneakily looked at before on a slip of paper, fold it before the Volunteer can read it, and give it to the Volunteer for safekeeping. Also give the Volunteer the remaining pile of 48 cards.

Now the Volunteer should look at each of the face-up cards, start with that number, and place cards face-down on each while counting up to 10. For example, if one of the cards is a 7 then the Volunteer would place 3 cards face-down on top, counting “8, 9, 10”. Picture cards count as 10 already, so no cards would be placed on top of those. Be sure that you can still see the original face-up cards.

When all four face-up cards have piles on them, have the Volunteer add the values of the face-up cards. For example, if the cards were a 7, a King, an ace, and a 4 then the total would be 7+10+1+4=22. The Volunteer should count off that many cards face-down from the top of the remaining deck, and then turn over the very last card (the 22^nd card, in the case above). Have the Volunteer then look at the folded paper that you wrote on earlier, and Lo and Behold you’ve predicted that very card! Woo hoo!

Like yesterday’s trick, this one is a variation from card-trick.com. Thanks site!

An Easy Card Trick for a shortened week

It’s back to school after a long weekend (and the start of school for many, like our resident now-kindergartener!) so here’s a card trick that is easy enough to do while you’re occupied with either the fading summer or the burgeoning fall.   (Plus if, like yours truly, you only get Right versus Left correct about 50% of the time, it’s still OK.)   I’ll post a fancier card-trick tomorrow.

For this trick, start with a deck with any number of cards in it, as long as that number is 1 (boring),  4, 9,  16  (good for teaching to kids), 25, 36, or 49 (fancy, but it takes a long time to lay out so you might be back to boring).      Show the cards to a Volunteer, and tell the Volunteer to choose one of them secretly and memorize it.  Then give the deck to said Volunteer to shuffle as many times as desired and to lay out face-up in a square.

Have the Volunteer point to the row that the Chosen card is in.  You then memorize the card on far LEFT of the row.  We’ll call this card the Identifier.

Now gather the cards into a pile as follows:  start with the card at the bottom left, and scoop up that column of cards (so the card at the bottom left is at the top of the pile).  Move (right) to the next column and scoop up bottom to top.  Continue until you have one pile of cards.  [Alternately, you could scoop down top to bottom.  Just be consistent and do the same thing for each column.]

Now lay the cards into a square row by row.  [You might notice that the cards that made up the left-hand column are now the top row.  Indeed, if you scooped top to bottom then you’ll just have a transposition of the original layout; scooping bottom to top masks that just a little bit.]   The car you memorized in the previous square, the Identifier, will be somewhere in that top row.  Have the Volunteer point again to the row that the Chosen card is in.  The Chosen card will be in the same column as the Identifier, making it easy to determine.  [If you lay out the rows bottom to top or a mixture of both — laying out a row and then doing  rows above and below in any order — the Identifier won’t be at the top, but still will be at in the same column as Chosen card.]

At this point you could simply state what the Chosen card is, or you could get all fancy and just memorize it, then gather the cards up, look at them, think about it, and then announce the Chosen card.  Hooray!

This trick is a variation of one found at card-trick.com.

Math Card Tricks: I Know the 5th Card

Ξ and I were recently shown a card trick by a colleague and a former student. It was a two-person show, with the student playing the role of “Assistant” and our colleague playing the “Mind Reader”. This is what we saw.

The Mind Reader left the room, and the Assistant explained that she could communicate with the Mind Reader telepathically. We chose five cards (total) from a standard deck of 52, then showed them to the Assistant. Here are our cards:

She picked one of the five (the ♥2), told us to remember it, hid it, then arranged the remaining four cards in a row, like so:

The Assistant then told the Mind Reader to enter (why didn’t she just “think” him back in?). The Mind Reader studied the cards and proudly proclaimed that the last card was the ♥2. Ta-da!

More Math Magic

Create a chart on a piece of paper like the one below. It could be any NxN square; there’s nothing special about the fact that N=7 in this example.

1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31	32	33	34	35
36	37	38	39	40	41	42
43	44	45	46	47	48	49

Give the chart to a Member of the Audience (or a student in your class, or your Great-Aunt Mabel, or…). Without you watching, this person should circle a number, cross off all the other numbers in that row and column, and then pass the paper to another person. The next person circles one of the remaining numbers, crosses off everything else in that number’s row and column, and passes it along. This continues until everything on the chart has either been circled or crossed off. At this point, the last person to circle a number adds up all the circled numbers and think very hard about the sum.

You, meanwhile, know that people are circling numbers but don’t know which numbers are being circled. Once the final person has added up the numbers and telepathically thinks of the sum, you will use your Amazing Mindreading Powers to announce the sum!

Which will be 175.

Or, in general for an NxN square, will be (N³+N)/2. Pretty cool, isn’t it? Of course, if you do this trick more than once for the same crowd, you might want to vary the size of the square to mask the fact that the same size square always leads to the same answer.

Here’s a brief justification for the total. Because numbers in the same row and column as a circled number are crossed out, it turns out exactly one number in each row is circled and, at the same time, exactly one number in each column is circled.

When you add, think of adding the numbers row by row.

• The first number will be between 1 and 7 (inclusive). Call it A, where 1≤A≤7.
• The second number will be between 8 and 14 (inclusive). Call it 7+B, where 1≤B≤7.
• The third number will be between 15 and 21 (inclusive). Call it 2·7+C, where 1≤C≤7.
• Continue all the way to the seventh number, which will be between 43 and 49. Call it 6·7+G, where 1≤G≤7.

When you add these all up you get (A)+(7+B)+(2·7+C)+…+(6·7+G). Reordering gives (7+2·7+…+6·7)+(A+B+…+G), which can be simplified as (1+2+…+6)·7+(A+B+…+G).

Because you don’t know which numbers were circled, you don’t know what A, B, etc. are. But you do know that one number was picked in each column! Each number in the first column is 1 more than a multiple of 7, each number in the second column is 2 more than a multiple of 7, etc.

Since one number is picked out of each column, the numbers A, B, …, G are 1, 2, …, 7 in some order. This means that A+B+…+G is just 1+2+…+7, and the total amount is (1+2+…+6)·7+(1+2+…+7). In general, for an NxN square, the total will be (1+2+…+[N-1])·N+(1+2+…+N). The first summation is (N-1)N/2 and the second summation is N(N+1)/2, so the total becomes {(N-1)N/2}·N+N(N+1)/2. Factoring out N/2 gives (N/2)(N²-N+(N+1)), which becomes N(N²+1)/2 or (N³+N)/2.

But of course, you don’t have to give away your secret.

I found this trick on curiousmath.

Math Card Tricks: Finding the selected card

Another card trick to amaze friends and family! For this you need an ordinary deck of 52 cards. (It is OK if it has more or fewer, but you do need to know exactly how many cards are in the deck.)

Shuffle the deck, and then have a Volunteer from the Audience think of a number between 1 and 20. Give the deck to The Volunteer, and tell them to remove that many cards from the deck while your back is turned. If it makes them feel better, they can shuffle the cards again afterwards. The Volunteer should put the removed cards out of sight and not tell you how many cards were taken, although it is fine for The Volunteer to tell the rest of the audience.

Mathematical Mindreading: Splitting up Piles

Here’s another mindreading trick with which to amaze your friends and family! Start with a pile of pennies (or beads or paper clips or poker chips etc.) You’ll need to know in advance how many pennies there are: for this example we’ll assume that there are 20 pennies in the pile. It’s not a big deal if your Audience knows that you know how many pennies there are, although if you do this trick more than once it is most impressive to use a different number of pennies each time.

Math Card Tricks: More Counting Cards

Looking for another card trick? This one only requires simple counting and moving cards around but it’s mostly done behind your back so it’s a little tricky (as it were).

Start with a Volunteer From The Audience. Tell The Volunteer to think of a number n from 1-15, and then, while your back is turned, to count down that many cards from the top of the face-down deck and memorize the n^th card. For example, if The Volunteer thinks of the number 3, then the card to be memorized is the 3^rd card from the top. The cards should remain in the same order — The Volunteer can look at the card, but doesn’t move anything around.

Card Tricks: The Year Game

Here’s a another math card trick. And by “math” I mean…well…counting. All the way to twelve. It’s really not that hard a trick.

Start by taking a deck of cards (if a few are missing it’s not a big deal) with the jokers at the bottom. Have a Volunteer From The Audience pick a card, memorize it, and put it back on top of the pile. Then hand the pile of cards to The Volunteer and let them cut the deck a few times. You can turn around or hide your eyes if you want to; it really doesn’t matter, but it looks a little more impressive.

Math Card Tricks: Counting Cards

Here’s a simple card trick that can be done with an ordinary deck of cards and a Volunteer From the Audience. Remove the jokers (from the deck, not the audience), and shuffle the cards.

Throughout this trick, Aces are 1, 2s are 2, 3s are 3, …, 10s are 10, Jacks are 11, Queens are 12, and Kings are 13.

A computer that really CAN read your mind…

Want to play another mind-reading game? Check out this Mindreader Applet by Anup Doshi. In this game you enter a supposedly random* sequence of digits 0 and 1 and the computer tries to predict your next move. The computer scores a point each time it correctly predicts your move; you score a point each time it doesn’t. Can you beat the computer? To get a sense of how the program is working, try moving in a predictable sequence (101001010010100) and see how long it takes for the computer to figure out the pattern.

* Click here for Scott Adams’ take on random number generation.

The original version of this game was created by Professors Yoav Freund and Rob Schapire, based on work of Claude E. Shannon and D. W. Hagelbarger in the 1950s.

Mind-Reading and Card Tricks: the Kruskal count

Want another mind-reading trick? This one uses an ordinary deck of cards and is very impressive, although it only works about 85% of the time.

Here’s how to play. Have a person think of a number n between 1 and 10 and keep that number secret. You shuffle a deck of cards, and then slowly deal the deck face up card by card as the person watches. The n^th card is a “key” card, in the words of Ivars Peterson, and the value of that card — counting aces as 1, face cards as 5, and all other cards as the number shown — tells the person how many cards ahead the next key card is. For example if the initial number is 7, then the 7^th card is the first key card. If that card is a Jack (worth 5) then the person counts ahead (as you continue to slowly deal) 5 more cards, and that 5^th card is the new key card. If that card is a 3, then the 3^rd card after that is the next key card, etc. All of this is done in silence.

Eventually the deck runs out. You look at the person and, using your amazing mind-reading/mathematical powers, you show them their final key card!

Math-free Mindreading!

I sensed (well, OK, I checked the blogstats) that the recent Math Magic posts on Mindreading were getting a lot of hits.   So for fun, here* is another mind-reading site.  In this one you are shown some cards, you pick one, and the computer uses amazing powers (mind-reading and/or listening skills) to determine which card you picked and remove it from the selection.  There’s no actual math in this one, although searching for it did reveal even more sites that do combine math and mind-reading, so there’s plenty of fodder for future entries on this topic!

* This site seems to have a lot of advertising on it, though I know I’ve seen less commercial sites with the same trick.