Mathematical Tricks and Treats

Steve Bergen ()

Overview: I started teaching math back in 1973 when I graduated from college. Over the years, I have loved teaching people mathmatical number tricks, especially for "mental multiplication." I have also been fascinated by various mathematical word problems for which the answer seems to run strikingly counter to intuition.

Former website of mathtutor33.com is now located at summercore33.com/mt -- click

T R I C K S

T R E A T S

Inspiration: at a recent Thanksgiving dinner hosted by Cheryl and Donald Fischer (incredible food, by the way), I was reacquainted with a cousin of Donald who now teaches 4th graders in NYC, Amanda Shyman. I shared a number of these tricks and treats with Amanda and decided it was time to put them together on a web page. Back in the 80s, I had created an Apple IIe software program called The TC Multiplier, so I decided on this Sunday morning to copy and paste the Apple IIe DATA statements into an HTML web page. Steve Bergen 11/28/99

if you or any reader finds mistakes or wants to suggest improvements or enhancements, I would be glad to correspond with you!
-- Steve Bergen (e-mail: )

p.s. No answers yet are posted to these treats. It is important to think about these treats for a while and to share ideas with friends, teachers and other students. If I posted answers, it would deprive you the of the tremendous joy of pondering these problems. Once you have thought about it for a while and shared thoughts with others, I would be glad to "check your answer" and respond to you via ; happy thinking .... Steve

Notes from readers

>Subject: Re: The next letter is...
>Date: Wed, 22 Feb 2006 20:12:17 -0500
>
>"Kimberly Martin" on Wednesday, February 22, 2006
>at 12:35 PM -0500 wrote:
> >N.  My reasoning is that the letters that contain any curves
> >(non-straight
> >lines) are omitted.
> >I enjoyed your Web site!
> >Kim
> >Massachusetts

I'm an accountant going back to school for teaching and I tutor on a part 
time basis.  Would you mind if I use that trick for my education class?
Not a problem!  Keep up the good site!  =o)

And another

Mr. Bergen,

I would very much like to figure out the answer to the finger snapping problem
I just can't remember how to do it.Ê I've done a problem very similar to this one
but i can't remember the formula that is used.Ê Do you think youÊcould help me out?
 
Thanks, Tyler K
solution as JPEG graphic

solution as EXCEL spreadsheet

And another

Pete Peters -- melisma90 AT hotmail.com -- 1/11/08 wrote: Forgive me if I am writing to where I shouldn't be writing but I came across your program on math tricks/problems etc and before I die [I'm 72] I would like to find how a trick my father taught me, many years ago, actually works. I've searched since I was about 20 for the answer. The trick was this :- A farmers house had a window in each of the 4 walls. Out of each window he could see 7 men working in the fields and didn't want to see any more than this. One day a man came looking for work but was refused as the farmer wouldn't have 8 men in one of the fields. The man said if I can arrange it so you will never see more than 7 men will you give me a job? The farmer gave him a job and indeed he never saw more than 7 in each field. This happened again & again with more men and even though about 10 or more men applied for jobs the farmer never could see more than 7 men in each field. My dad taught me this and I managed to do it a few times but as the years flew by the procedure flew by as well. If you have the answer to this I would be very grateful to receive it. Thanks, Pete Peters

And another

Undisclosed-Recipient AT cox.net on June 1, 2008 at 1:08 PM -0400 wrote:

Steve:
The newspaper presented an interesting math problem this morning.
It said a man from Dallas had paid $6 for his $3 parking ticket in
Wisconsin. Twenty-one years late.

But the article also said:
A notice on the ticket indicated that failure to pay within 120 hours
would result in doubling of the $3 forfeiture.

I thought it would be fun to figure out what he really owed, if you get
my drift. It's similar to that old double a grain of rice over the 64
squares of a chessboard problem.Ê As I have it figured, you need to
double $3 1,533 times.
 
I found your finger snap problem and thought I'd let Excel do the hard
work the same way (instead of me pushing x2 on a calculator that many times.

The number gets so big that Excel refuses to play after a while.
How else do I find the answer, in the computer age?
Steve

P.S. Excel quit at the number 1.3483E+308

What is this number? 1.3483 X 2.71828183 followed by 308 zeros?

Steve:
I'm a lawyer in Phoenix, Arizona.
I don't have much occasion for math in my line of work.
But my calculus professor in college had to throw out my test scores because they skewed his curve so radically.
(I leave to you to speculate on which end of the curve.) (When I pull out those tests now, I have no clue what I'm looking at.)
I also used to substitute teach fourth graders in the Washington School District here in Phoenix.
And I'm thinking of teaching 4th graders full-time as an "encore career" in another 10 years or so.
I enjoyed your webpage.
Steve

hello Steve ... I met a math teacher in California this summer and shared your problem with him ... here is his response hope this helps ... Mathematica is a high powered software program .... my friend confirmed that there was no easy trick in Excel ... Steve

Hi Steve, OK, so there are 1,533 120-hour periods in 21 years. The value we want, then, is 3 x 2^1533. When I tried it in Excel, I got the same result as your email correspondent. Excel wouldn't compute past 3 x 2^1022, which is 1.2483 x 10^308. So I turned to Mathematica, which can compute the value of 3 x 2^1533 with ease. The answer is: 903867160095387220717543727260617792342034967979633356759878139649471521846926200250906884040254768623793287355333001795448406367048109371863604870459040494295705002268109459289043279219130562378817078676520432521599004696714224666048020170787315486874180865503362754111052508230606535654422811487919701388888363415958096978645337004336034418780217345444087872643084606303742486693068148926827515693347689842622314212014739129697191246650462370071376785127243776. In scientific notation, this is 9.039 x 10^461.


Take two cups -- one with coffee and one with tea -- of the exact same size, let us say 8 ounces. Put one tablespoon of the coffee into the tea. Now take one tablespoon from the tea cup (which is mostly tea, but a little bit of coffee) and put it back into the coffee cup. Please note that both cups have plenty of "head space" so that nothing overflows. Will there be more coffee in the tea cup or more tea in the coffee cup? Or will they be the same?

Circulated by Dan Meyer 3/3/11 ... Blog Entry and Three Acts of Videos

Let C1 = coffee in mug 1 after the 2 swaps
Let T1 = tea in mug 1 after the 2 swaps
Let C2 = coffee in mug 2 after the 2 swaps
Let T2 = tea in mug 2 after the 2 swaps
We know C1+T1=8 because after 2 swaps mug 1 has 8 ounces of liquid
We know that C1+C2=8 because they represent all the coffee in the 2 mugs
Therefore C1+T1 = C1+C2 (transitive)
And now we subtract C1 from each side, getting
T1=C2 (QED)
James Stuart Tanton is a mathematician and math educator, born in Austrailia. He is the founder of The St. Mark’s Math Institute, scholar at the Mathematical Association of America, author of over ten books on mathematics. Here is his excellent youtube video


On Saturday night, 1/10/2004 when the New England Patriots beat the Tennessee Titans, a frozen football with ice was found weighing 6 pounds plus half its weight. How much did it weigh? (suggested by Jon Latson RI)


This is a classic problem (mathmatical hall of fame) from the book series TC Mits and TC Wits. Tie an imaginary belt (perhaps with rope) around the equator of the earth. Let us assume it is 25,000 miles. Make this belt so tight so that you cannot fit even a pencil or a pin under it, let alone a finger. Now add 10 feet of rope to this belt, so that it is 25,000 miles plus 10 feet. Pretend that you can distribute the rope evenly around the earth so that it is uniformly "loose" around the entire earth. How loose will the new belt be? Would a dog be able to walk under it? Could I stick my fist under it? What about a finger or a pen or pin?


At what age will a baby's height in inches equal his/her weight in pounds. The average height of a baby at birth is around 19" and the average weight is about 7 pounds. This means that height in inches is more than double weight in pounds. But by the time the child gets to be a young teenager, he or she is generally 4 or 5 feet tall which is of course 48" or 64"; in contrast, all teenagers weigh much more than 48 or 64 pounds. So what is the EXACT AGE at which a baby's height in inches equals weight in pounds?

Click here for a picture that suggests that this special day is beyond 36 months! Not sure if this is true!


Snap your fingers. Wait 1 second. Snap again. Wait 2 seconds. Snap again. Wait 4 seconds. Snap again. Each time, your "wait" will be double the previous time. Do this for one year. How many times will you have snapped your finger?

This is a neat problem that comes from Ask Marilyn in Parade Magazine January 16th 2005 and was submitted by Michael Storey from Kingston WA: "At 60 mph, it takes 60 seconds to travel a mile. At 120 mph, it takes 30 seconds. At what speed would it take 45 seconds?" I will re-phrase the question in terms of race car drivers Alice, Barbara and Carol:

--> Race car driver Alice drives at 60 mph and takes 60 seconds to travel one mile.

--> Race car driver Barbara is much faster and drives at 120 mph; it takes Barara 30 seconds to travel one mile.

--> Race car drive Carol has a broken speedometer but travels one mile in exactly 45 seconds. What was her speed?

This neat sequence of letters comes from a colleague in California (Kyle Krupnick) who like me is fascinated with puzzles and whose youngest daughter used this Web page to mastered most of these amazing math tricks ... he writes that "I have never seen her so excited" and poses the letter sequence of A E F H I K L M and challenges readers to guess the next letter!  

CHOP THE NUMBER IN HALF. IF THE RESULT IS EVEN, JUST ATTACH A FINAL ZERO. IF THE RESULT IS ODD, THEN FORGET ABOUT THE REMAINDER AND JUST ATTACH A FINAL FIVE. THIS TRICK WORKS BECAUSE MULTIPLYING BY 5 GETS THE SAME RESULT AS MULTIPLYING BY 1/2 x 10. EXAMPLE: 5 x 468 = 2340 ANOTHER EXAMPLE: 5 x 467 = 2335

ALTHOUGH MOST PEOPLE JUST MEMORIZE THE EASY SQUARES, AN INTERESTING PATTERN IS THAT THEY INCREASE BY JUMPS THAT MATCH THE ODD NUMBERS. CONSIDER ... 1,4,9,16,25,36,49,64,81,100 ... THE GAPS BETWEEN THEM ARE 3,5,7,9,11 !

HERE IS A GREAT TRICK FOR SQUARING A TWO DIGIT NUMBER THAT ENDS IN 5: TAKE THE FIRST DIGIT AND MULTIPLY IT BY THE NUMBER THAT FOLLOWS IT IN THE 'NUMBER ALPHABET'. THEN ATTACH A 25 AND YOU ARE DONE. FOR EXAMPLE, TRY 75 X 75. SINCE 7 X 8 = 56, THE ANSWER WILL BE 5625 !

THE TOUGHER SQUARES CONTINUE THE SAME PATTERN OF ODD NUMBER JUMPS THAT THE EASY SQUARES STARTED: 121,144,169,196, 225,289,324,361,400,441,484,529,576,625 NOTICE THE JUMPS OF 23,25,27, ETC. ALSO, NOTICE THAT 12^2 = 11^2 +11+12. THIS RECURSIVE PATTERN ALWAYS EXISTS!

WHEN YOU MULTIPLY 11 BY A 2 DIGIT NUMBER, AND THE SUM OF THE 2 DIGITS IS BIGGER THAN 9, THEN YOU MUST MAKE THE SUM FLY DOWN ON TOP OF THE PATTERN, "WITH 1 # LANDING IN THE HOLE AND 1 # ON" TOP OF THE LEFT NUMBER, BUMPING IT BY 1, WHICH IS REALLY A CARRY. CONSIDER 48 X 11 ... WE HAVE 12 LANDING ON 4-8 "... THE 2 FLIES INTO THE HOLE ... 528 !

THIS TRICK JUST EXTENDS THE 11 TRICK, EXCEPT YOU STUTTER ONCE WITH THE NUMBER IN THE HOLE. FOR EXAMPLE, SINCE 11 X 35 = 385, THEN WE WILL HAVE 111 X 35 = 3885, WHERE THE 8 JUST GOT REPEATED AN EXTRA TIME. AND WHAT DO" YOU THINK THE TRICK FOR 1111 WILL BE ? VOILA ... 1111 X 63 = 69993 !

TO CHECK YOUR ANSWER, WHEN YOU MULTIPLY BY 3, THE DIGITS OF THE ANSWER WILL ALWAYS ADD UP TO BE A MULTIPLE OF 3. FOR EXAMPLE, 3 X 9 = 27 AND 2 + 7 = 9. OR TRY 3 X 56. THE ANSWER IS 168, SO ADD 1 + 6 + 8 TO GET 15, WHICH IS A" MULTIPLE OF 3. AND SO IT CHECKS!

ONE WAY TO MULTIPLY BY 4 IS TO DOUBLE THE NUMBER TWICE--THIS WORKS BECAUSE 4 EQUALS 2 X 2 !"

IF YOU WANT, YOU CAN DOUBLE THE NUMBER THREE TIMES, SINCE 8 = 2 X 2 X 2 ! FOR EXAMPLE, CONSIDER 8 X 7. TAKE THE 7 AND DOUBLE IT. YOU GET 14. DOUBLE 14 TO GET 28. THEN DOUBLE IT ONE LAST TIME TO GET 56. YOU'RE DONE -- 7 X 8 = 56. 8,16,24,32,40,48,56,64,72,80 ...

WHEN MULTIPLYING 9 BY ANY NUMBER > 1 & < 10, YOU WILL GET A 2 DIGIT ANSWER." DIGIT ONE WILL BE 1 LESS THAN THE NUMBER TO BE MULTIPLIED AND YOU CAN GET DIGIT TWO BY SUBTRACTING DIGIT ONE FROM 9. CONSIDER 9 X 8. 1 LESS THAN 8 IS 7." 7 FROM 9 IS 2, SO THE ANSWER IS 72." 9,18,27,36,45,54,63,72,81,90 ..."

TREAT THE NUMBER IN THE TEENS AS A SUM OF A SINGLE DIGIT AND A MULTIPLE OF 10. THEN GET 2 ANSWERS AND ADD ... FOR EXAMPLE: 7 X 16 = 7 X (10 + 6) =" 70 + 42 = 112 !

WHEN YOU MULTIPLY 11 BY A 2 DIGIT NUMBER, PUSH THE NUMBER APART, YIELDING A 3 DIGIT NUMBER WITH A HOLE IN THE MIDDLE. THEN, IN THE MIDDLE, PUT SOME NUMBER, THAT IS SUM NUMBER, THAT IS THE SUM OF THE TWO OUTSIDE NUMBERS GOES IN THE MIDDLE. FOR EXAMPLE, 11 X 34 = 3?4 WHICH GIVES US 374 !

STEP 1: ADD EITHER NUMBER TO THE ONE'S PLACE OF THE OTHER NUMBER. STEP 2: MULTIPLY THE STEP 1 ANS. BY 10. STEP 3: MULTIPLY THE ONE'S PLACES OF THE ORIGINAL TWO NUMBERS. STEP 4: ADD STEP 2 TO STEP 3. VOILA ! FOR EXAMPLE: 13 X 18 ... 21 ... 210 3x8 = 24 now add 210 + 24 to get 234


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