# Transcript: Counting on an Answer | Mar 26, 1999

A title reads “TVO’s Virtual Classroom. Get connected.” The “V” in “Virtual” is a tick, the “A” in “classroom” is an “at sign” with an extended loop that turns into a power cord with a plug at the end, and the first “O” in “classroom” is a spinning globe.(Lively music plays)

In front of a world Atlas spread on the wall, Mister C., curly-haired and bearded in his forties, with glasses, wearing a white T-shirt with painted-on suspenders and a drawing of a horse on it, says

HI, AND WELCOME TO

THE VIRTUAL CLASSROOM.

WELCOME 1999 AND LESS

THAN A YEAR TO GO BEFORE

THE FIRST CELEBRATION

OF THE MILLENNIUM.

NOW, THAT'S AN INTERESTING

QUESTION BY ITSELF.

IS THE NEW YEAR'S EVE IN

360-SOME-ODD DAYS, IS THAT THE

REAL BEGINNING OF THE

MILLENNIUM OR NOT, AND CAN YOU

EXPLAIN THAT MATHEMATICALLY?

ANYWAY, I'M GOING TO

LEAVE THAT ONE WITH YOU.

BEFORE THE HOLIDAY, I DID

LEAVE YOU WITH A BIT OF A

PROBLEM TO DO WITH, NOT

CUTTING A PIECE OF CAKE,

WHICH EVERYBODY WAS

QUITE WELL WITH.

On a sheet of paper featuring a single dice turned up on number 5, a drawing of a box-shaped object shows it divided into 27 smaller similarly shaped objects by 4 vertical and 2 horizontal cuts. Mister C. appears in a small frame at the bottom right of the screen, and his hand is shown writing “less than 6 cuts” on the paper.

Mister C. continues IT WAS TO DO WITH TAKING A

CUBE AND CUTTING IT INTO 27

SMALL CUBES, AND THE QUESTION

WAS, COULD YOU CUT THIS BIG

CUBE INTO 27 SEPARATE PIECES

WITH LESS THAN SIX CUTS?

NOW, WHAT I WOULD LIKE IS

SOMEBODY TO PHONE IN AND LET

ME KNOW WHETHER IT'S POSSIBLE

TO DO IT WITH LESS THAN

SIX CUTS OR NOT, AND

EITHER WHY OR WHY NOT.

SO I WOULD CERTAINLY

INVITE YOU TO CALL IN.

REMEMBER, IT'S POUND NINE

AND WE CAN TAKE YOUR CALLS

STARTING NOW.

YOU KNOW, IF I DON'T SEE ANY

CALLS REALLY QUICKLY, I JUST

MIGHT CALL OUT AND SEE IF I

CAN GET SOMEBODY TO RESPOND

THAT WAY.

SO PLEASE CALL IN IF

YOU HAVE SOME IDEA.

WELL, I THINK I'M GOING TO TRY

RYAN ARMSTRONG FROM CAYUGA

AND SEE IF HE'S THOUGHT ABOUT

THE PROBLEM AND, IF NOT,

MAYBE I CAN WORK MY WAY

THROUGH IT WITH HIM.

SO IN A MOMENT OR TWO,

WE'LL BE CONNECTED TO RYAN.

WHEN YOU HEAR THE PHONE

RING, RYAN, PICK IT UP.

HAVE YOU...

ARE YOU THERE, RYAN?

Ryan says YEAH.

Mister C. continues HI, RYAN.

DID YOU HAVE A CHANCE TO THINK

ABOUT THIS LITTLE PROBLEM

OVER THE HOLIDAY?

Ryan says NO.

Mister C. continues OKAY, WELL, HERE'S

THE PROBLEM AGAIN.

YOU BASICALLY HAVE A BIG CUBE,

AND WHAT YOU'RE DOING IS

YOU'RE TRYING TO CUT IT INTO

27 PIECES, AND MY QUESTION

WAS, CAN IT BE DONE

WITH LESS THAN SIX CUTS?

NOW, I JUST WANT YOU TO GO

WITH A SORT OF, I'LL CALL IT

A GUT FEELING.

DO YOU THINK YOU CAN DO IT

WITH LESS THAN SIX CUTS OR NOT?

Ryan says I DON'T KNOW.

Mister C. continues WHAT DO YOU...

YOU KNOW, I'M JUST

ASKING FOR A SENSE.

WHICH WAY WOULD YOU LEAN

IF YOU HAD TO MAKE A BET

ON THE SPOT?

SIX CUTS...

Ryan says I CAN'T HEAR YOU

FROM THIS CORNER.

Mister C. continues HMM?

DIDN'T QUITE HEAR THAT.

OKAY.

Ryan says I DON'T UNDERSTAND

THE QUESTION.

TRY SOMEONE ELSE.

Mister C. continues OKAY.

I'D BE HAPPY TO

TRY SOMEBODY ELSE.

LET'S SEE.

LET'S TRY... SEAN.

HELLO, AM I

SPEAKING TO SEAN?

Sean says HELLO?

Mister C. continues HI, IS IT SEAN?

Sean says YEAH.

Mister C. continues WHAT DO YOU THINK?

DO YOU THINK YOU COULD DO

THIS IN LESS THAN SIX CUTS?

Sean says PARDON ME?

Mister C. continues DO YOU THINK YOU COULD CUT A

CUBE, MAKE 27 SMALL ONES WITH

LESS THAN SIX CUTS?

WHAT DO YOU THINK?

ALL I'M LOOKING

FOR IS A FEELING.

I KNOW YOU CAN'T THINK

THROUGH THE WHOLE PROBLEM.

Sean says I THINK SO, YEAH.

Mister C. continues OKAY.

NOW, I WANT YOU

TO TAKE A REAL...

I PUT THIS DIE ON

HERE FOR A REASON.

WHAT IS THE EXACT SHAPE OF THE

PIECE THAT'S IN THE MIDDLE

OF THE 27?

Sean says A SQUARE.

Mister C. continues IT'S GOING TO BE A CUBE,

JUST LIKE THIS, ISN'T IT?

Sean says YEAH.

Mister C. continues NOW, IF I'M GOING TO MAKE A

CUBE, A SINGLE CUBE AND I HAD

A PIECE OF LUMBER AND I HAD

TO CUT IT DOWN TO SIZE,

HOW MANY CUTS WOULD

I HAVE TO MAKE?

ONE FOR EACH...?

Sean says YEAH, ONE FOR EACH.

Mister C. continues FACE.

HOW MANY FACES ON A CUBE?

Sean says SIX.

Mister C. continues SO MY ARGUMENT, IF YOU FOLLOW,

IS SINCE THE MIDDLE PIECE IN

HERE IS A CUBE, THEN IT TAKES

AT LEAST SIX CUTS TO MAKE IT;

THEREFORE, YOU CAN'T DO

IT WITH LESS THAN SIX.

Sean says NO.

Mister C. continues OKAY.

THANKS FOR HELPING ME OUT, AND

THAT'S ALL I WAS AFTER, AND I

CERTAINLY APPRECIATE THAT.

OKAY, WHAT I'M GOING TO DO

RIGHT NOW IS I ALWAYS START

WITH A PROBLEM OF SOME SORT.

THAT WAS TAKING UP A PROBLEM.

I'M GOING TO GO AFTER

THIS ONE RIGHT NOW.

A rectangular crossword-type grid pattern has nine symmetrically placed shaded squares, three rows with three blank spaces in each, separated by mathematical symbols, and three similar columns. The totals of each row, represented by equal signs followed by numbers, read 9, 10 and 31. The totals of each column read 3, 8 and 2. Loose cut-outs are printed with numbers from 0 to 8. Mister C's hand moves the cut-outs to the blank spaces in the grid.

Mister C. continues IT'S A LITTLE NUMBER PROBLEM

AND I GET MY NUMBERS SORT OF

OUT HERE SO YOU CAN SEE THEM.

THERE ARE NINE DIGITS HERE,

STARTING WITH ZERO, RUNNING UP

TO EIGHT, AND THERE ARE NINE

BLANK SPACES ON THIS GRID.

ONE, TWO, THREE, FOUR, FIVE,

SIX, SEVEN, EIGHT, NINE.

NOW, ALL THE COLUMNS AND THE

ROWS BASICALLY CALCULATE TO A

TOTAL WHICH IS SHOWN AT THE

END, SO IN OTHER WORDS,

THIS NUMBER TIMES THIS

NUMBER DIVIDED BY THIS

HAS TO EQUAL THREE.

IF YOU'RE WORKING DOWN HERE,

THIS NUMBER TIMES THIS NUMBER,

WHATEVER THAT TOTAL IS,

PLUS SOME OTHER NUMBER HAS

TO EQUAL EIGHT.

LIKEWISE GOING DOWN HERE, THIS

NUMBER, SUBTRACT THAT NUMBER

DIVIDED BY THIS HAS TO BE TWO.

GOING ACROSS, THREE

NUMBERS TO ADD TO NINE.

THIS NUMBER DIVIDED BY THIS

NUMBER TIMES THIS NUMBER HAS

TO BE TEN.

THIS NUMBER TIMES THIS NUMBER,

SUBTRACT THIS HAS TO BE 31.

NOW, THERE ARE A COUPLE OF

THINGS THAT ARE A LITTLE BIT

OF A CLUE IN HERE, AND

THE WAY I WANT TO TAKE THIS UP

WITH YOU IS SORT OF ONE STEP

AT A TIME.

He appears in a small frame at the bottom right of the screen.

Mister C. continues I'M GOING TO PROPOSE THAT THIS

ROW IN THE CENTRE IS WHERE YOU

SHOULD START BECAUSE THERE'S

ONE DIGIT THAT HAS TO BE

PLACED IN THAT ROW SOMEWHERE.

AND IF YOU CAN THINK ABOUT IT

FOR A MOMENT OR TWO, REMEMBER

WHAT YOU WANT IS A TOTAL OF

TEN AND WHAT YOU'RE DOING IS

ONLY DIVIDING AND MULTIPLYING.

SO I'M GOING TO INVITE YOU TO

PHONE IN RIGHT NOW AND LET ME

KNOW WHAT DIGIT WOULD HAVE TO

GO IN THAT ROW FOR SURE AND SO

WE CAN GET THE PUZZLE STARTED.

NOBODY HAVE ANY IDEAS?

WELL, LET'S SEE.

LET'S TRY KEN.

SO IN A MOMENT OR TWO, I'LL

SEE IF WE CAN WORK OUR WAY

THROUGH IT STARTING WITH KEN.

SO WHEN YOU HIT...

HELLO, KEN.

Ken says HI.

Mister C. continues OKAY, I'M GOING TO SORT

OF COACH YOU THROUGH THIS

JUST TO GET THIS STARTED.

NOW, I'M GOING BACK

TO THAT MIDDLE ROW.

WHAT ONE OF THESE DIGITS

HAS TO APPEAR THERE?

WHAT DO YOU THINK?

Ken says FIVE.

Mister C. continues YOU WANT TO BET.

HEY, NOW WHERE...

WHICH ONE OF THE THREE SLOTS

IS LIKELY TO HAVE THAT FIVE?

Ken says THE MIDDLE.

Mister C. continues THE MIDDLE.

YOU SURE ABOUT THAT?

Ken says NO.

Mister C. continues SOMETHING DIVIDED BY FIVE,

WELL, IS THERE ANYTHING THAT'S

DIVISIBLE BY FIVE

OTHER THAN ZERO HERE?

REMEMBER, ALL THESE HAVE TO BE

WHOLE NUMBERS WHEN YOU GET

TO THE END.

Ken says OKAY, THE FIVE SHOULD

BE IN THERE, YEAH.

Mister C. continues OKAY, I KIND OF HELPED

YOU A LITTLE BIT.

He places the number 5 cut-out in the third blank space of the second row.

Mister C. continues NOW, WHAT DO YOU KNOW ABOUT

THE RESULT OF THIS DIVISION?

WHAT NUMBER DOES

IT HAVE TO BE?

Ken says TWO?

Mister C. continues IT HAS TO BE A TWO, YEAH.

SO THERE ARE SOME

POSSIBILITIES HERE,

AREN'T THERE?

THERE'S MORE THAN ONE

POSSIBILITY, LIKE FOR

INSTANCE, EIGHT

DIVIDED BY FOUR, RIGHT?

Ken says YEAH.

Mister C. continues OKAY.

NOW, I'M GOING TO LEAVE IT

AT THAT UNLESS YOU HAVE A

FEELING YOU KNOW WHICH PAIR.

REMEMBER, IT COULD BE EIGHT

DIVIDED BY FOUR, SIX DIVIDED

BY THREE OR FOUR

DIVIDED BY TWO.

Ken says OKAY.

Mister C. continues OR EVEN TWO DIVIDED BY

ONE IS POSSIBLE AS WELL.

I DIDN'T SEE THAT ONE.

Ken says OKAY.

Mister C. continues DO YOU HAVE A FEELING?

Ken says NO.

Mister C. continues OKAY.

NOW, I'M GOING TO GET YOU

JUST TO TAKE A LOOK MAYBE

AT ANOTHER SPOT.

DO YOU HAVE ANY SENSE OF

ANOTHER NUMBER YOU MIGHT

BE ABLE TO FILL IN?

Ken says UM, THE... NO.

Mister C. continues OKAY.

WANT YOU TO LOOK AT

THIS NUMBER UP HERE.

Ken says THE SIX.

OR NOT.

Mister C. continues IT COULD BE A SIX.

SIX SUBTRACT FIVE IS ONE,

DIVIDED BY SOMETHING EQUALS

TWO, BUT I DON'T THINK

WE CAN DO THAT, CAN WE?

SO SIX ISN'T IT.

YOU KNOW THAT IT'S GOT TO BE

BIGGER, SO WHAT ABOUT

THIS NUMBER?

Ken says SEVEN.

Mister C. puts the 7 cut-out in the first blank space of the third column.

Mister C. continues SEVEN.

IF YOU HAD SEVEN DIVIDED BY

FIVE, OR EXCUSE ME, SEVEN

SUBTRACT FIVE, WHAT DOES THIS

NUMBER HAVE TO BE DOWN HERE?

Ken says SEVEN... ZERO.

Mister C. continues NOT DIVIDED BY ZERO.

YOU'RE IN THE

RIGHT IDEA, THOUGH.

Ken says ONE?

Mister C. puts the 1 cut-out in the third blank space of the third column.

Mister C. continues YEAH.

OKAY, THANK YOU VERY MUCH, AND

YOU'VE COMPLETED A COLUMN.

AND I'M GOING TO TRY TO GET

SOMEBODY TO FINISH IT UP FOR ME.

SO IF ANYBODY ELSE WOULD LIKE

TO PHONE IN, I'LL GIVE YOU

A CHANCE FIRST.

OTHERWISE, I'M GOING TO...

LET'S TAKE A LOOK AND

SEE WHO WE HAVE HERE.

I'M GOING TO GO TO MADONNA

THIS TIME AND MICHELLE,

I'M PHONING YOU RIGHT NOW.

IN A MOMENT OR TWO, WE'LL...

REMEMBER, MICHELLE, WHEN YOU

HEAR THE PHONE RING, PICK IT UP.

HELLO, MICHELLE?

Michelle says HELLO?

Mister C. continues ARE YOU THERE?

HI, MICHELLE.

She says YEAH, HI.

Mister C. continues NOW, YOU'VE SEEN

SOME OF THESE NUMBERS.

DO YOU THINK YOU CAN HELP ME

OUT WITH FINISHING OUT PART

OF THE PUZZLE ANYWAY?

I MIGHT DRAW YOUR

ATTENTION TO UP HERE.

She says THERE'S A PROBLEM OVER HERE

AT MADONNA, SO I CAN'T...

Mister C. continues TURN THE TV DOWN.

I CAN HEAR THE FEEDBACK.

JUST A LITTLE BIT.

IS THAT HELPING?

She says WE DON'T HAVE A PICTURE.

Mister C. continues YOU DON'T HAVE --

OH, OKAY, YOU HAVEN'T

GOT A PICTURE.

MAYBE THE FACILITATOR

COULD PHONE THE HELP LINE?

ANYWAY, WITHOUT A PICTURE, I

KNOW YOU CAN'T HELP ME, SO

THANK YOU VERY MUCH FOR GOING

ALONG AS FAR AS YOU COULD GO.

I'M GOING TO TRY

NICOLE AT CAYUGA THEN.

SO, REMEMBER, WHEN THE

PHONE RINGS, PICK IT UP,

AND I HOPE...

HELLO, IS IT NICOLE?

Nicole says YEAH.

Mister C. continues OKAY, LET'S LOOK AT THESE TWO

SPOTS UP HERE AND THE NUMBERS

THAT ARE LEFT OVER.

WHAT TWO NUMBERS DO YOU

THINK THEY HAVE TO BE?

She says ONE AND TWO.

Mister C. continues NOT ONE AND TWO, BUT

TWO'S ONE OF THEM.

THE OTHER ONE HAS TO BE...?

ZERO, RIGHT?

I'VE ALREADY USED THE ONE UP.

SO THE ONLY TWO I CAN PUT

THERE TO ADD UP TO NINE ARE

ZERO AND TWO, RIGHT?

She says YEAH.

Mister C. continues OKAY.

NOW, THE ONLY QUESTION IS,

I GUESS IS WHAT SPOT DO

I PUT THEM IN?

WHAT DO YOU THINK?

I COULD EITHER PUT THE TWO

HERE AND THE ZERO THERE OR THE

OTHER WAY AROUND.

She says THE OTHER WAY AROUND.

OKAY, LET'S CHECK IT OUT.

NOW, LET'S SEE.

IF WE'RE TRYING TO MAKE A

THREE DOWN HERE AT THE BOTTOM,

THIS NUMBER'S GOING TO...

IT DOESN'T MATTER

WHAT THE NUMBER IS.

WHAT IS ZERO TIMES THE NUMBER?

She says IT'S NOT POSSIBLE.

Mister C. continues WHAT'S ZERO TIMES ANY NUMBER?

She says ZERO.

Mister C. continues RIGHT.

ZERO DIVIDED BY ANY

NUMBER IS GOING TO BE?

She says ZERO.

Mister C. continues SO I GUESS THE ZERO'S GOT TO

GO IN THE OTHER SPOT, RIGHT?

She says YEAH.

Mister C. continues AND THAT'S HOW YOU DO

THESE KINDS OF PUZZLES WITH

THAT KIND OF LOGIC. OKAY.

SO LET'S SEE IF WE CAN

COMPLETE THIS ONE.

NOW, REMEMBER WHAT

NUMBERS WE HAVE LEFT.

WE GOT THE THREE, THE FOUR,

THE SIX AND THE EIGHT.

TWO TIMES SOMETHING, SO

TWO TIMES THREE IS SIX...

BUT THAT'S NOT GOING

TO WORK, IS IT?

She says NO.

Mister C. continues WHAT NUMBER DO YOU THINK

MIGHT WORK IN THIS SLOT

RIGHT HERE?

She says TWO TIMES SIX.

Mister C. continues RIGHT.

She says DIVIDED BY THREE.

Mister C. continues DIVIDED BY...?

She says FOUR.

Mister C. continues FOUR.

I LIKE THAT.

WELL, LET'S FINISH

UP THIS OTHER ONE.

DOES IT WORK?

She says YEAH.

Mister C. continues WHERE DO I PUT THE

OTHER TWO NUMBERS?

She says ZERO TIMES THREE.

Mister C. continues RIGHT ON.

AND THAT WORKS BOTH WAYS.

SIX DIVIDED BY THREE IS

TWO, TIMES FIVE IS 10.

THANK YOU VERY MUCH.

He completes the columns, which read downward “2 times 6 divided by 4 equals 3. 0 times 3 plus 8 equals 8, and 7 minus 5 divided by 1 equals 2.” he removes the grid.

Mister C. continues I LIKE DOING PUZZLES LIKE

THAT, AND ONE OF THE REASONS

THAT I REALLY LIKE TO TAKE THE

TIME TO DO THOSE IS THAT I

THINK IT'S REALLY IMPORTANT

THAT YOU LOOK AT THE LOGIC OF

HOW NUMBERS WORK TOGETHER,

THAT IF YOU WANT A 10 AND

YOU'RE MULTIPLYING, YOU NEED

A FIVE, ESPECIALLY IF YOU'RE

TALKING ABOUT WHOLE NUMBERS.

YOU GOT TO LOOK AT

DIVISIBILITY, WHETHER

SOMETHING DIVIDES BY SOMETHING

TO GIVE A WHOLE NUMBER OR NOT,

AND IT'S IMPORTANT.

THE MORE FACILITY YOU HAVE

WITH WORKING WITH NUMBERS LIKE

THAT IN THAT PUZZLE, THE

EASIER YOUR MATHEMATICS WILL

BE LATER ON.

NOW, IT JUST SO HAPPENS

THAT I HAVE MY PAL, MARC,

BACK IN THE STUDIO.

WE'RE GOING TO HAVE A LITTLE BIT

OF A CHAT IN A MOMENT OR TWO.

WHAT I'M GOING TO SAY IS THAT

NEXT WEEK, WE WILL TALK A BIT

ABOUT THE MYSTERY BECAUSE IT

IS OUR LAST WEEK, BUT I KNOW

THAT MARC'S BEEN UP TO

NO GOOD SO I'M GOING

TO TALK TO HIM RIGHT NOW.

A life-sized red-haired and moustachioed rag doll wearing a gray pinstriped coat and a white cap appears beside him, looking at some papers on the desk.

Mister C. says HI, MARC.

HOW ARE YOU DOING?

YOU KNOW, SINCE THE LAST TIME

I TALKED TO YOU, THE WEATHER'S

BEEN JUST ABSOLUTELY

UNBELIEVABLE, HASN'T IT?

Inside a white outlined frame on a blackboard, a caption reads “Marc Replies - Must be global colding!”

Mister C. continues GIVE ME A BREAK.

A LITTLE BIT OF SNOW AND YOU'VE

BECOME TOTALLY DISCOMBOBULATED.

ANYWAY, I UNDERSTAND THAT

YOU'VE BEEN TALKING TO A

PROFESSOR FROM U. OF T. ABOUT

THE DONNY GROVER CASE.

Inside a white outlined frame on a blackboard, a caption reads “Marc Replies - Yep. You're right on both counts.”

Mister C. continues YOU KNOW, U. OF T. IS AN

ABSOLUTELY OUTSTANDING

UNIVERSITY, BUT I'VE NEVER

HAD A CHANCE TO VISIT IT.

YOU WOULDN'T HAPPEN TO HAVE

ANY PICTURES, WOULD YOU?

Inside a white outlined frame on a blackboard, a caption reads “Marc Replies - it just so happens I have some video footage. Here it is...”

A clip shows the University of Toronto with its older and more modern buildings. A large porticoed and domed building appears.

Mister C. continues OH, THAT'S A... THOSE

ARE OLDER BUILDINGS.

I DIDN'T REALIZE IT.

I WOULD HAVE IMAGINED THAT IT

WAS QUITE A MODERN UNIVERSITY.

AH, AN INTERESTING MIX

IN ARCHITECTURE AS WELL.

I HEAR THAT THAT'S

CONVOCATION HALL.

THERE ARE A NUMBER OF

CONCERTS THAT ARE HELD THERE.

[birds chirping]

WELL, JUST WHY WERE YOU IN

TOUCH WITH A U OF T PROFESSOR?

Inside a white outlined frame on a blackboard, a caption reads “Marc Replies - Doctor Ned Barleau, Professor of Mathematics, is a “combinatorics” expert.”

Mister C. continues A WHAT?

Inside a white outlined frame on a blackboard, a caption reads “Marc Replies - He specializes in counting problems. He took all of those pairs of code numbers to try to look for patterns.”

Mister C. continues DID HE FIND A PATTERN?

Inside a white outlined frame on a blackboard, a caption reads “Marc Replies - Not yet! He told us to come back next week.”

Mister C. continues HEY, I THINK THAT

THAT'S A GOOD IDEA.

NEXT WEEK WE'LL

WORK ON THE MYSTERY

AND WE'LL COME BACK TO IT.

GOODBYE, MARC.

WE'LL SEE YOU LATER.

Turning toward the camera, he says ANYWAY, I WILL INDEED TIE UP

THE LOOSE ENDS ON THE MYSTERY

SO AT LEAST YOU KNOW WHAT

THAT STORY WAS ALL ABOUT

AND SEE HOW IT FITS INTO THE

MATHEMATICS WE'VE BEEN TRYING

TO DO FOR THE

LAST LITTLE WHILE.

NOW, WHAT YOU SHOULD HAVE

BEEN DOING OVER THE CHRISTMAS

HOLIDAYS WERE TWO EXERCISES.

BOTH OF THEM WERE

FAIRLY EXTENSIVE.

IF YOU WERE WORKING WITH A

PARTNER OR HAD THE OPPORTUNITY

TO WORK WITH A PARTNER, IT

WOULD HAVE BEEN A LOT EASIER,

I SUSPECT.

WHAT I'M GOING TO DO IS TAKE

UP A FEW SELECTIONS FROM BOTH

EXERCISE NUMBER EIGHT

AND EXERCISE NINE.

I CAN'T DO EVERY SINGLE

QUESTION, BUT I CAN TALK ABOUT

A FEW OF THEM, AND I'M GOING TO

START WITH THIS ONE RIGHT HERE.

A blue sheet of paper titled “Sequences” shows five illegible columns of figures.

Mister C. continues WE'RE GOING TO DO

THESE SEQUENCES.

NOW, WHAT I'D LIKE YOU TO DO

IS TO PHONE IN AND GIVE ME A

LITTLE BIT OF HELP ANYWAY.

I WOULD LIKE TO FIND OUT,

FIRST OF ALL, WHAT THE BLANKS

ARE AND THEN, SECOND OF ALL,

ONCE I'VE GOT THE BLANKS,

THEN I WANT TO DEAL WITH WHAT

THE RULE IS, AND I DO HAVE

A COUPLE OF QUESTIONS I WILL ASK

YOU ABOUT THE RULES LATER ON.

SO I INVITE YOU TO PHONE

IN RIGHT NOW AND TELL ME,

WHAT ARE THE NUMBERS THAT WOULD

FILL IN THESE TWO SPOTS HERE?

WE HAVE A CALL, GOOD.

IN A MOMENT OR TWO,

WE'LL BE CONNECTED

TO EITHER MATT OR JOHN.

JUST A SECOND.

THERE WE ARE.

IS IT MATT OR JOHN?

A voice says JOHN.

Mister C. continues OKAY, JOHN, WHAT

ARE THESE TWO PARTS?

WHAT SHOULD GO IN

EACH ONE OF THEM?

John says THIRTY-TWO AND SEVEN.

Mister C., writing in the figures, continues THIRTY-TWO OVER

SEVEN, THANK YOU.

John says AND 64 AND EIGHT.

Mister C. continues OKAY.

NOW, WHAT'S GOING

ON IN THE NUMERATOR?

John says THE TOP...

THE NUMERATOR'S

DOUBLING EACH TIME.

Mister C. continues DOUBLING EACH TIME, AND THE

BOTTOM'S PRETTY OBVIOUS,

ISN'T IT?

IT'S JUST A NORMAL,

TWO, THREE, FOUR, FIVE.

IT'S JUST THE NUMBERS, RIGHT?

John says YEAH.

Mister C. continues NOW, I'M GOING TO ASK YOU

WHILE YOU'RE ON THE PHONE,

I'D SORT OF LIKE YOU TO DO

THE ONE BELOW IT AS WELL.

DO YOU KNOW WHAT THE

ANSWER IS TO THIS ONE?

John says THE ONE BELOW IT.

Mister C. continues THE ONE RIGHT BELOW IT.

John says THE NEXT TWO NUMBERS

ARE 32 AND 7.

Mister C. continues YEAH.

John says AND 64 AND 8.

Mister C. continues YEAH, NOW, I WANT...

I'M GOING TO MAKE

A MINOR CHANGE.

YOU'RE ABSOLUTELY RIGHT,

BUT THAT'S EQUAL TO 8.

BUT HOW DID YOU

KNOW THAT SEQUENCE?

IT DOESN'T LOOK

RIGHT, DOES IT? ONE, TWO, ONE, EIGHT,

EIGHT, THIRTY TWO - HOW DID YOU GET THAT?

WHAT DID YOU REALIZE?

John says THAT THE TOP NUMBERS

ARE KIND OF THE SAME.

IT'S ONE, TWO AND THREE.

THEN IT'S FOUR, FOUR, THEN

IT'S EIGHT AND FIVE,

THEN IT'S EIGHT.

LIKE, THE TWO PAIRS OF

NUMBERS ARE THE SAME.

Mister C. continues THESE TWO SEQUENCES

ARE IDENTICAL.

THE ONLY DIFFERENCE IS THAT

I REDUCED THIS ONE, RIGHT?

John says YEAH.

Mister C. continues AND THAT'S WHAT I --

I WAS SORT OF HOPING SOMEBODY

WOULD SEE THAT AND YOU'VE DONE

A REALLY NICE JOB.

THANK YOU VERY MUCH.

John says ALRIGHT.

Mister C. continues OKAY, WE'RE GOING TO GO ON

TO ANOTHER COUPLE HERE.

GET THIS OUT OF THE WAY.

AND, AGAIN, I'D LIKE YOU TO

PHONE IN AND LET ME KNOW

HOW TO COMPLETE THESE

TWO SEQUENCES.

Another blue sheet titled “Sequences” shows two rows of figures that read “1, 1, 2, 3, 5, 8, 13, __ and 1, 8, 11, 69, __, 96, 101.”

Mister C. continues OKAY, HAVE WE GOT MATT?

A male voice says NO, MATT KIND OF

WENT TO THE BATHROOM.

Mister C. continues OKAY, WHO AM I SPEAKING TO?

The voice says OKAY.

Mister C. continues WHO IS IT?

The voice says IT'S JAMIE.

Mister C. continues OKAY, JAMIE.

WHAT'S THE NEXT BLANK?

WHAT'S THE NEXT NUMBER

IN THIS SEQUENCE?

Jamie says 18.

Mister C. writes the figure in the end space

of the first row and continues 18!

HOW DO YOU CALCULATE 18?

HOW DID YOU GET THAT?

Jamie says TIMES FOUR.

Mister C. continues NOW, I WANT YOU TO GO BACK

THROUGH THIS, LIKE, YOU GOT A

DIFFERENCE OF FIVE HERE,

RIGHT, WHICH IS...

Jamie says OH, OH, ADD ONE AND TWO, AND

THEN THREE AND THEN FOUR

AND THERE YOU GO.

Mister C. continues OH, I SEE WHAT YOU DID.

Jamie says YEAH.

Mister C. continues DOES THAT WORK FOR

ALL THE NUMBERS?

DOES THAT RULE THAT YOU USE

WORK FOR EVERY NUMBER

THAT'S IN THE SEQUENCE?

Jamie says PARDON ME?

Mister C. continues DOES THE NUMBER THAT YOU USE

WORK FOR EVERY SINGLE NUMBER

IN THE SEQUENCE?

OR THE RULE THAT YOU USED?

Jamie says OH.

Mister C. continues THAT'S REALLY KEY BECAUSE IF

IT WORKS FOR A COUPLE OF THEM,

THAT'S GREAT, BUT IT'S GOT

TO WORK FOR ABSOLUTELY

EVERY ONE OF THEM.

I THINK YOU'RE CLOSE

TO THE RIGHT IDEA,

BUT I DON'T THINK

YOU HIT IT.

Jamie says OKAY.

Mister C. continues OKAY?

Jamie says YEAH.

Mister C. continues IS THERE ANOTHER PERSON OUT

THERE THAT MIGHT HAVE ANOTHER

IDEA ABOUT THIS PARTICULAR

SEQUENCE WITH A DIFFERENT

ANSWER HERE?

OKAY, LET'S SEE...

I'M GOING TO TRY CARA.

SO IN A MOMENT OR TWO, CARA,

WHEN YOU HEAR THE PHONE RING,

PICK UP THE RECEIVER, PLEASE.

HELLO.

A voice says HELLO?

Mister C. continues CARA, ARE YOU THERE?

Cara says YEAH.

Mister C. continues OKAY, DO YOU HAVE A DIFFERENT

ANSWER FOR THIS ONE?

Cara says 21.

Mister C. crosses out 18 and substitutes 21.

Mister C. continues OKAY, EXPLAIN

WHAT YOUR RULE IS.

HOW DID YOU GET 21?

WHAT'D YOU DO?

Cara says I FORGET.

IT WAS A LONG TIME

AGO WHEN I DID THIS.

Mister C. continues BUT YOU'VE GOT THE RIGHT

ANSWER, SO I MEAN,

ALL WE GOT TO DO IS GO BACK IN

THE MEMORY BANKS AND FIGURE

OUT WHAT'S HAPPENING THERE.

LOOK AT THIS NUMBER (he points to 2).

HOW DOES IT RELATE TO

THE NUMBERS BEFORE?

Cara says OH, I KNOW HOW I DID IT.

Mister C. continues OKAY.

Cara says YOU ADD THE FIRST NUMBER

TO THE SECOND NUMBER.

Mister C. continues AND YOU GET...

IF YOU ADD THESE TWO,

YOU GET THIS ONE, RIGHT?

HOW DO YOU GET

THIS NUMBER HERE?

Cara says TWO PLUS THREE EQUALS FIVE.

Mister C. continues OKAY, YEAH, YOU'RE GOING

AHEAD TO THIS NUMBER HERE,

BUT THAT'S...

TWO PLUS THREE IS FIVE.

HOW DO YOU GET THIS NUMBER?

Cara says THREE PLUS FIVE.

Mister C. continues RIGHT.

IN OTHER WORDS, IT'S THE

ADDITION OF THE TWO NUMBERS

PREVIOUS, RIGHT?

Cara says YEAH.

Mister C. continues SO 8 PLUS 13 IS 21.

WELL DONE.

THANK YOU VERY MUCH.

Cara says OKAY, BYE.

Mister C. continues I APPRECIATE YOUR HELP.

OKAY, LET'S LOOK AT THIS ONE.

He points to the lower sequence.

Mister C. continues THIS IS ONE OF MY FAVOURITE

SEQUENCES, BUT IT'S NOT

STRAIGHTFORWARD NECESSARILY.

SO IS THERE ANYBODY THAT HAS

AN IDEA WHAT THE ANSWER IS?

SO I'M LOOKING FOR A NUMBER

THAT'S SOMEWHAT IN THE MIDDLE.

LET'S SEE...

OKAY, LET'S TRY SOMEBODY ELSE.

LET'S TRY RYAN.

SO, RYAN, WHEN YOU HEAR THE

PHONE RING, OF COURSE, PICK IT

UP AND WE'LL TRY TO

FIGURE THIS ONE OUT.

JUST A SECOND OR TWO.

I'M SURE WE'LL BE CONNECTED.

HELLO, RYAN, ARE YOU THERE?

Ryan says HELLO?

Mister C. continues HAVE YOU GOT ANY IDEA

HOW THIS SEQUENCE WORKS?

Ryan says OUT OF THE WHOLE PAPER, THAT

WAS THE ONLY ONE I DIDN'T GET.

Mister C. continues OKAY.

NOW STAY WITH ME.

YOU CAN SEE THE SCREEN

CLEARLY, I PRESUME.

Ryan says MM-HMM.

Mister C. continues OKAY, WATCH WHAT I DO RIGHT

NOW AND SEE IF THIS IS

A BIT OF A CLUE.

He turns the sheet of paper upside down

Mister C. continues LOOK AT THAT SAME

SEQUENCE AGAIN.

NOW, ONE LITTLE HINT

HERE IS THESE ONES.

IF THEY WERE...

THOSE LITTLE THINGS ON THE

ONE SHOULDN'T BE THERE (he erases the tags on the number ones).

WHAT'S TRUE ABOUT...

WHEN I HAD IT THIS WAY AND

WHEN I TURNED IT OVER?

WHAT ABOUT EACH ONE OF THE

NUMBERS, ALTHOUGH THE

ORDER'S DIFFERENT?

Ryan says IT'S THE SAME UPSIDE DOWN?

Mister C. continues OKAY, EXACTLY.

SO THIS SEQUENCE HAD SOMETHING

TO DO WITH, WELL, MAYBE

SYMMETRY OR WHATEVER

YOU WANT TO CALL IT.

NOW THAT YOU'VE SEEN THAT

LITTLE BIT OF A TRICK, ANY

IDEA WHAT THE MISSING NUMBER

IS BETWEEN 69 AND 96?

Ryan says 88?

Mister C. continues YOU GOT IT.

THANK YOU VERY MUCH.

WELL DONE.

OKAY, WE'LL GO

TO A COUPLE MORE.

NOW, THIS ONE...

A sheet of paper titled “Sequences” has two rows of numbers that read “1, 4, 9, 16, __ and 1, 3, 7, 15, 31, __.”

Mister C. continues I WOULD ARGUE THAT THAT'S

ONE THAT ABSOLUTELY

EVERYBODY SHOULD HAVE GOTTEN.

AND WHAT I'M GOING TO DO AT

THIS POINT IS I'M GOING TO DO

A QUESTION AND INSTEAD OF

HAVING SOMEBODY PHONE IN,

I'LL GET YOU ALL TO

ANSWER THE QUESTION.

SO THERE IT IS.

A blue sheet of paper titled “Question Number 1” appears, and he reads the text. The term numbers read 1, 4, 9 and 16. The multiple choice answers read “1. 2n - 2. n+3 - 3. N “squared.” - 4. None of the the above.”

Mister C. continues THE GENERAL TERM, “N.”

IS THE TERM NUMBER.

REMEMBER THAT WHAT I'M TALKING

ABOUT WHEN I SAY “TERM

NUMBER,” TERM NUMBER ONE, TWO,

THREE, FOUR, SO TERM NUMBER

ONE IS 1, TERM NUMBER

TWO IS 4, TERM NUMBER

THREE IS 9 AND SO ON.

SO WHAT I'M ASKING YOU HERE

IS WHAT IS THE FORMULA,

OR THE GENERAL TERM FOR THE

SEQUENCE, 1, 4, 9, 16?

IF YOU THINK IT IS 2n, PRESS 1,

IF YOU THINK IT IS N+3, PRESS 2,

IF YOU THINK IT'S N SQUARED,

I COULDN'T DO A PROPER

EXPONENT HERE PARTLY

BECAUSE THE SOFTWARE

WON'T HANDLE THAT.

AND IF YOU DON'T THINK IT'S ANY OF THE ONES ABOVE.

NOW, I SEE 50 PERCENT

HAVE ANSWERED.

I THINK THAT WE'RE GOING TO

HAVE TO TAKE THE QUESTION AS

IT IS RIGHT NOW AND WE'LL TAKE

A LOOK AT WHAT YOUR ANSWERS ARE.

A coloured bar graph appears with most answers (4 and 3) in second and fourth place.

Mister C. continues OKAY, THAT'S WHAT YOU'VE SAID.

SO FOUR OF YOU THOUGHT

IT WAS N PLUS 3.

ONE PERSON THOUGHT IT WAS N

SQUARED, AND A NUMBER OF PEOPLE

THOUGHT IT WAS

NONE OF THE ABOVE.

NOW, LET'S GO BACK TO THE

ACTUAL QUESTION, 1, 4, 9, 16.

WHAT'S THE NEXT NUMBER

IN THIS SEQUENCE?

PLEASE PHONE IN.

OKAY, SO WE GOT SOMEBODY

CALLING IN, GOOD.

SO WHAT WE WANT TO KNOW IS

WHAT THE NEXT NUMBER IS AND

HOW IT WAS CALCULATED.

AND IN A MOMENT, WE'LL EITHER

BE TALKING TO JOHN OR MATT.

HELLO, IS IT JOHN OR MATT?

John says IT'S JOHN.

Mister C. continues OKAY, JOHN, WHAT'S

THE NEXT NUMBER?

John says 25.

Mister C. writes it in and continues OKAY, WHAT ARE YOU DOING

EACH TIME YOU GET

ANOTHER NUMBER HERE?

IN OTHER WORDS, WHAT'S

THE ONE AFTER 25?

John says THE ONE AFTER 25 IS 36.

Mister C. continues SURE.

YOU KNOW EXACTLY

WHAT YOU'RE DOING.

SO WHAT IS THE GENERAL TERM?

WHAT'S HAPPENING EACH...

IF THIS IS TERM NUMBER ONE AND

THAT'S TERM NUMBER TWO AND

THAT'S TERM NUMBER THREE, THAT

FOUR, THAT FIVE AND SIX,

HOW ARE YOU ACTUALLY

CALCULATING THIS IF YOU

KNOW THE TERM NUMBER?

John says GEE, I FORGOT.

Mister C. continues PARDON?

John says I FORGOT.

Mister C. continues WELL, WHAT'S THE RELATIONSHIP

BETWEEN ONE AND ONE, TWO AND

FOUR, THREE AND

NINE, FOUR AND 16?

THERE'S A COMMONALITY HERE.

John says OH, THEY...

EACH OF THEM TIMES

THEMSELVES EQUALS THE NEXT.

Mister C. continues EXACTLY.

SO YOU'VE GIVEN ME THE ANSWER.

NOW, HOW DO I WRITE

THAT IN GENERAL FORM?

IF THIS IS THE N TERM,

WHAT IS THE ACTUAL NUMBER?

IT'S... YOU TOLD ME.

IT'S THE NUMBER

MULTIPLIED BY ITSELF.

HOW DO YOU WRITE THAT?

John says LIKE, N SQUARED.

Mister C. continues RIGHT, EXACTLY.

SO THE ANSWER IS N SQUARED.

THAT WAS ANSWER NUMBER THREE.

THANK YOU VERY MUCH.

LET'S GO TO THE ONE BELOW IT,

AND, ONE MORE TIME, I WOULD

SORT OF LIKE TO HEAR WHAT YOUR

ANSWER IS, WHAT'S THE NEXT

VALUE IN THIS

PARTICULAR SEQUENCE? OKAY.

AND IN A MOMENT OR TWO, WE'LL

BE CONNECTED, I PRESUME, TO...

HELLO, IS IT MATT?

A voice says HELLO?

Mister C. continues HELLO, IS IT MATT

OR JOHN AGAIN?

John says JOHN.

OKAY, MATT WANTS

TO TALK TO YOU.

OKAY, NEVER MIND.

Mister C. continues OKAY.

OKAY, SO WE'RE NOW

MOVING TO THIS ONE.

WHAT'S THE NEXT VALUE HERE?

IS IT MATT?

Matt says 63.

OKAY, GREAT, MATT.

ANY IDEA WHAT

THE RULE IS HERE?

HOW WOULD YOU DESCRIBE HOW

YOU GET THE NEXT NUMBER?

Matt says THE LAST NUMBER PLUS ITSELF.

LIKE, DOUBLE EACH NUMBER.

Mister C. continues DOUBLE EACH NUMBER PLUS...?

Matt says OH, PLUS ONE.

Mister C. continues YEAH, OKAY.

SO THAT'LL DEFINITELY WORK.

I'M GOING TO SPEAK TO CARA OR

HANNAH AND SEE IF THEY HAVE

SOMETHING ELSE TO

ADD ON THIS ONE.

SO EITHER...

I'M CONNECTING TO EITHER...

HELLO, IS IT

CARA OR HANNAH?

Hannah says IT'S HANNAH.

Mister C. continues -OKAY, WHAT DID YOU SEE HERE?

HAVE YOU GOT SOMETHING TO

ADD TO WHAT WAS SAID BEFORE?

She says I'M SORRY, I CAN'T

HEAR YOU VERY WELL.

Mister C. continues OKAY, THIS PARTICULAR

SEQUENCE, YOU AGREE WITH 63?

LET'S START THERE.

She says 127.

Mister C. continues YEAH, OKAY, 127'S NEXT.

NOW, HOW DID YOU

CALCULATE THAT?

WHAT WAS YOUR

APPROACH TO DOING IT?

She says PARDON?

Mister C. continues HOW DID YOU CALCULATE THAT?

She says EIGHT TIMES TWO PLUS ONE.

Mister C. continues OKAY.

SO BASICALLY,

EVERYBODY'S DOING THAT.

BUT I WANT TO SHOW YOU

SOMETHING AND, OF COURSE,

I WANT TO WRITE ON AN ANGLE

SO I CAN'T QUITE DO THAT.

BUT IF YOU LOOK AT

THIS, THIS IS...

IF YOU LOOK AT THIS SET OF

NUMBERS RIGHT BELOW IT, NOTICE

THAT EACH AND EVERY TIME, THE

NUMBER THAT I HAVE BELOW IT,

IF I ADD ONE TO ALL OF THEM

IS 2, 4, 8, 16, 32, 64.

THOSE ARE DOUBLED EACH TIME,

AND IF THIS IS TERM NUMBER ONE

AND THAT'S TERM TWO AND THAT'S

TERM THREE AND THAT'S TERM

FOUR, THERE'S ANOTHER WAY

TO GET A FORMULA HERE.

TO GET THIS NUMBER -- LET'S

START WITH THAT ONE --

IT'S THE SAME AS TWO TO THE

EXPONENT ONE MINUS ONE,

SO THAT'S THAT ONE THERE.

THIS NUMBER I COULD CALCULATE

BY TWO TO THE EXPONENT

TWO MINUS ONE.

THIS ONE IS TWO TO THE

EXPONENT THREE MINUS ONE.

THAT'S THE KIND OF FORMULA

THAT I WAS REALLY HOPING

THAT YOU MIGHT DISCOVER.

WHAT YOU FOUND WHEN YOU DID IT

YOUR WAY, DOUBLE AND ADD ONE,

IS SOMETHING CALLED A

RECURSIVE, WHICH IS OKAY

EXCEPT THAT IF YOU WANT THE

117th TERM, YOU GOT TO FIGURE

OUT THE OTHER 116

BEFORE YOU GET IT.

BUT WITH THIS ONE, THEN YOU

CAN JUST USE THE EXPONENT.

SO THAT'S JUST A LITTLE TIP.

ANYBODY GET THIS ONE?

A blue “Sequence” sheet reads O, T, T, F, F, S, S, __

IF NOT, I'M GOING TO

LEAVE IT WITH YOU.

AH, WE GOT AN ANSWER. OKAY.

AND IN A MOMENT OR TWO, WE'LL

BE CONNECTED TO PROBABLY JOHN.

MUST BE THE WEATHER.

EVERYTHING'S

SLOWER THESE DAYS.

HELLO, IS IT JOHN?

John says YEAH.

Mister C. continues OKAY, WHAT'S THE LETTER

THAT GOES IN HERE?

John says E.

Mister C. continues WHY?

John says BECAUSE IT GOES 1, 2,

3, 4, 5, 6, 7, 8.

Mister C. continues 6, 7, 8.

RIGHT ON. THANKS, JOHN.

OKAY, I WANT TO

DO YET ONE MORE.

WHOOPS, YES, THAT'S

THE NEXT QUESTION.

A blue sheet of paper titled “Question Number 2.” It reads “The general term (“n” is the term number) for the sequence 5, 11, 17, 23, ... is: with 4 multiple choice options that read “6n - 1, 6n + 5, 3n + 2 and None of the above.”

Mister C. continues I WANT YOU TO DO

THIS QUESTION HERE.

SO I WANT YOU TO LOOK PRETTY

CLOSELY AT THESE AND SEE

IF YOU CAN GET THE

CORRECT VALUE.

AND ONCE WE GET OVER 50

PERCENT, I'LL BE HAPPY.

WE HAD THAT THE LAST TIME.

ONE OR TWO MORE ANSWERS

AND WE'VE GOT IT.

OKAY, LET'S TAKE A LOOK

AT WHAT YOU RESPONDED.

The coloured bar graph shows 5 answers in 1, 3 in 2, 1 in 3 and 1 in 4.

Mister C. continues WHAT WE HAVE HERE IS

AN INTERESTING SPLIT.

WE HAVE MOST PEOPLE THINKING

IT'S 6N MINUS 1, THREE

PEOPLE THINKING 6N PLUS

5 AND JUST A COUPLE OF

PEOPLE WITH THE OTHER FORMULA.

NOW, I'M GOING TO GO

BACK TO THIS SEQUENCE.

He writes the figures on a sheet of paper, showing the difference of 6 between them above.

Mister C. continues THE SEQUENCE THAT

YOU HAD WAS 5, 11, 17 and 23.

EACH TIME, THERE'S A

DIFFERENCE OF 6.

NOW, THAT 6 IS REALLY,

REALLY, REALLY IMPORTANT

BECAUSE WHEN YOU'RE WORKING

OUT THE FORMULA, IF I HAVE

THIS AS TERM NUMBER ONE AND

TERM NUMBER TWO AND TERM

NUMBER THREE AND TERM NUMBER

FOUR, IF I TAKE 6 TIMES 1,

I HAVE TO SUBTRACT

1 TO GET 5.

IF I TAKE 6 TIMES 2 AND

I SUBTRACT 1, I'LL GET 11.

IF I TAKE 6 TIMES 3, SUBTRACT

1, I GET THE NEXT NUMBER.

SO THE FORMULA HAS TO BE

6 TIMES THE TERM NUMBER,

SUBTRACT 1, SO THAT THE

CORRECT ANSWER IS 6N MINUS 1.

IF YOU ANSWERED THAT,

YOU DID A NICE JOB.

NOW, IN A SENSE, I'M GOING TO

SHORTCUT ON THAT PARTICULAR

EXERCISE BECAUSE, IN FACT,

WHAT WE JUST DID WITH THAT ONE

QUESTION WAS THE KIND OF

INVESTIGATION THAT I WANTED

YOU TO TRY TO WORK OUT.

IF YOU LOOK AT THE COMMON

DIFFERENCES AND WORK OUT WITH

RESPECT TO EACH TERM NUMBER

WHAT THE CORRECT VALUE SHOULD

BE, THEN YOU CAN EVENTUALLY

GENERALIZE TO THE GENERAL

TERM, OF COURSE.

ANYWAY, I WILL TAKE A QUESTION

AND THEN I'LL GO ON FROM THERE.

I THINK MATT IS

PHONING FROM CAYUGA.

JUST A MOMENT, WE'LL BE

CONNECTED AND I'LL GET MYSELF

SET UP HERE TO DO OUR

NEXT LITTLE PIECE.

SO, MATT, IN A MOMENT OR

TWO, I HOPE WE'VE GOT

OUR CONNECTION.

NONETHELESS...

AH, THERE WE ARE.

HI, MATT.

IS IT MATT?

A voice says I DIDN'T PRESS 8.

Mister C. continues HELLO?

The voice says IT WAS MINE.

Mister C. continues HELLO?

The voice says HELLO?

Mister C. continues THE BACKGROUND, I CAN HEAR

WHATEVER'S GOING ON IN THE

BACKGROUND, SO WHO

AM I SPEAKING TO?

Ken says THIS IS KEN.

Mister C. continues OKAY, KEN.

DID YOU HAVE A QUESTION?

Ken says NO.

Mister C. continues OKAY.

I'M GOING TO GO ON FROM THERE,

AND I'M GOING TO TALK A LITTLE

BIT ABOUT THE PROBABILITY

EXPERIMENTS THAT YOU WERE

DOING OVER THE HOLIDAYS,

IF YOU HAD A CHANCE.

THE FIRST EXPERIMENT THAT I

HAD INVOLVED SIX STRINGS, AND

I'VE GOT A FEW OF THEM WITH

ME, JUST TO GIVE YOU A SENSE

OF WHAT THAT PROBLEM WAS, AND

WHAT YOU WERE TO DO WAS TO

CONNECT THEM IN PAIRS AND

HOLD THEM IN A FIST

SOMETHING LIKE THIS.

AND THEN YOU WERE SUPPOSED TO

GET SOMEBODY ELSE WHO WOULD

RANDOMLY PICK PAIRS OF THEM

AND TIE THEM TOGETHER AT THE

BOTTOM, AND THE WHOLE POINT OF

THIS EXPERIMENT IS TO SEE WHAT

KIND OF A CONFIGURATION WAS

THE MOST LIKELY ONCE YOU LET

GO OF THEM AND TOOK A LOOK

AT THE STRINGS AFTERWARDS.

SO WHAT I'M GOING TO ASK YOU,

FIRST OF ALL, IS IF SOMEBODY

ACTUALLY CONDUCTED OR WORKED

ON THE EXPERIMENT AND CAME TO

SOME CONCLUSIONS ABOUT

WHAT DIFFERENT WAYS

COULD THOSE STRINGS TURN OUT,

WHAT KIND OF LOOPS COULD

YOU GET AT THE END.

SO HAVE WE GOT ANYBODY

THAT ACTUALLY WAS ABLE TO

DISCOVER THAT?

I WILL WORK ON THE ASSUMPTION

THAT WE HAVEN'T HAD A CHANCE

TO WORK ON IT, SO I'M GOING TO

JUST DO A QUICK DEMONSTRATION

UNDER THE GRAPHICS CAMERA.

THESE STRINGS HAVE BEEN SET UP

IN THE SAME WAY AS THE ONES

I HAD IN MY HAND.

THE ONLY DIFFERENCE HERE IS

THAT I'VE ACTUALLY PUT SOME

LETTERS AT THE BOTTOM.

He lays the strings out on a blue table. Each one has a lettered tag stuck at the bottom.

Mister C. continues NOW, REMEMBER, IT'S SUPPOSED

TO BE RANDOM, SO THAT'S THE

ONE SET OF STRINGS AND WE CALL

THAT C AND D AT THE BOTTOM.

HERE'S ANOTHER SET OF STRINGS,

AND YOU CAN SEE THAT THEY'RE

TIED AT THE TOP.

AND THEN THE LAST SET OF

STRINGS WITH A AND B ON IT.

NOW, AS IT TURNS OUT, LET'S

JUST TAKE THE A AS OUR

BEGINNING POINT.

IF I RANDOMLY CONNECTED A

WITH F, IT IS POSSIBLE THAT B

WOULD, IN FACT, CONNECT WITH

E AND YOU'RE ONLY LEFT WITH C

CONNECTING WITH D.

SO WHAT DO YOU END UP

WITH IN THAT SITUATION?

WHAT YOU HAVE IS A SMALL LOOP,

OR A RELATIVELY SMALL LOOP.

THIS ACTUALLY BECOMES

A MEDIUM SIZE LOOP

IF YOU SPREAD IT ALL OUT.

SEE THE CONNECTIONS THERE?

SO ONE POSSIBILITY IS THAT

YOU GET A SMALL LOOP AND A

MEDIUM SIZE LOOP.

LET'S GO BACK AND SEE IF

THERE'S ANOTHER POSSIBILITY.

AND ONE OF THE THINGS IN

MATHEMATICS THAT YOU DO HAVE

TO DO IS TO THINK LIKE THIS,

IN SOMETIMES A RATHER

ABSTRACT WAY, WHAT

ARE THE POSSIBILITIES?

B COULD CONNECT TO A.

THAT'S ENTIRELY POSSIBLE.

IT'S ENTIRELY POSSIBLE THAT

E WOULD CONNECT TO F

AND C WOULD CONNECT TO D.

NOW, I'M GOING TO ASK YOU A

QUESTION IN A MOMENT OR TWO,

SO I WANT YOU TO KEEP

WATCHING CLOSELY.

SO WHAT I ENDED UP HERE

WITH IS THREE SMALL LOOPS.

NOW, THERE'S ONE OTHER

POSSIBILITY IN A GENERAL SENSE.

LET'S SAY E CONNECTED WITH

A AND F CONNECTED WITH D

AND B CONNECTED WITH C.

IF YOU SPREAD THIS OUT,

YOU'D BEGIN TO REALIZE THAT

WHAT YOU HAVE IS

ONE HUGE LOOP.

SO THERE ARE THREE

POSSIBILITIES.

I'LL GET MY CHART BACK HERE.

AND TRY TO GET IT

LINED UP PROPERLY.

The chart reads “6 strings - 1 Big Loop, 1 Medium and 1 Small Loop, 3 Small Loops.”

Mister C. continues THE THREE POSSIBILITIES

WITH THE SIX STRINGS ARE -

He reads off the chart.

Mister C. continues WHAT I'D LIKE YOU TO DO IS TO

PHONE IN AND LET ME KNOW WHICH

OF THESE IS THE MOST LIKELY

TO HAPPEN AND WHICH ONE IS

THE LEAST LIKELY.

IS IT MOST LIKELY YOU'RE GOING

TO GET A BIG LOOP OR THREE

SMALL ONES OR THIS

ONE, OR VICE VERSA?

OKAY, WE HAVE A CALL AND

I THINK IT WILL BE KEN.

AND IN A MOMENT, WE WILL BE

CONNECTED, WITH A LITTLE BIT

OF LUCK.

OKAY, KEN?

Ken says YEAH.

Mister C. continues WHICH OF THESE THREE

POSSIBILITIES DO YOU THINK IS

THE MOST LIKELY?

Ken says THE MEDIUM AND THE SMALL.

Mister C. continues MEDIUM AND SMALL, OKAY.

WHAT ONE IS THE LEAST LIKELY?

Ken says THE ONE BIG ONE.

Mister C. continues THE ONE BIG ONE.

SO YOU THINK THAT THIS

IS THE LEAST LIKELY.

Ken says OR, NO, THE THREE LOOPS.

Mister C. continues THE THREE SMALL ONES.

OKAY, I'M QUITE HAPPY

TO CHANGE THAT FOR YOU.

NOW, I'M GOING TO ASK YOU WHY

YOU THINK THAT THAT ONE'S

THE LEAST LIKELY.

TELL ME WHAT YOU THINK.

Ken says WHY DO I THINK THAT'S

THE LEAST LIKELY?

Mister C. continues YEAH.

Ken says BECAUSE THEY...

IT WOULD TAKE...

I DON'T KNOW.

I JUST... I DON'T KNOW.

Mister C. continues THINK ABOUT THOSE LETTERS

AT THE BOTTOM OF THE LOOPS.

REMEMBER, I HAD A AND B FOR

THE ONE PAIR AND C AND D FOR

THE SECOND AND E AND F.

HOW MANY WAYS CAN I ACTUALLY

CREATE THOSE THREE SMALL LOOPS?

A HAS TO CONNECT

TO B, DOESN'T IT?

Ken says YEAH.

Mister C. continues AND C HAS

TO CONNECT TO D.

IT CAN'T MAKE A MISTAKE AND

CONNECT TO E OR F, RIGHT?

Ken says YEAH.

Mister C. continues SO WHAT I'M SAYING THERE IS

THERE'S ONLY REALLY ONE WAY

THAT THAT HAPPENS.

A ACTUALLY GETS CONNECTED TO

B, C ACTUALLY GETS CONNECTED

TO D AND E TO F, SO YOU'RE

RIGHT ABOUT THIS ONE.

Ken says OKAY.

Mister C. continues SO YOU'RE RIGHT ABOUT THE

THREE SMALL LOOPS BEING THE

LEAST LIKELY TO HAPPEN.

NOW, I'M GOING TO

TELL YOU SOMETHING.

Ken says THAT IT'S THE...

THE ONE BIG ONE IS THE

MOST LIKELY TO GET.

Mister C. continues AS IT TURNS OUT, YOU'RE

CORRECT NOW, THAT,

IN FACT, THAT IS THE MOST.

NOW, THE CATCH IS -- AND I

REALLY SHOULD LEAVE IT WITH

YOU -- IS WHY IS THIS ONE A

LITTLE BIT MORE LIKELY

THAN THIS ONE?

Ken says BECAUSE THERE IS

MORE COMBINATIONS.

Mister C. continues YOU'RE ACTUALLY RIGHT.

NOW, I GUESS THE QUESTION IS,

HOW MANY COMBINATIONS WILL

GIVE YOU A BIG LOOP AND

HOW MANY THE SMALL LOOP?

HOW WOULD YOU TRY

TO FIND THAT OUT?

LIKE, I'M NOT SUGGESTING THAT

YOU CAN DO IT RIGHT NOW AND ON

THE SPOT, BUT WHAT WOULD YOU

DO TO TRY TO FIGURE OUT THE

COMBINATIONS FOR

THIS AND THIS?

Ken says SIT DOWN AND FIGURE...

I DON'T KNOW.

Mister C. continues LIST THEM ALL OUT.

JUST LIST THEM OUT, YOU KNOW,

USING A, B, Cs AND Ds AND SO ON.

IF YOU DID IT REALLY

CAREFULLY, YOU'D FIND OUT

THE ANSWER.

OKAY?

Ken says OKAY.

Mister C. continues THANK YOU VERY

MUCH FOR YOUR HELP.

AS I SAY, THIS IS A GREAT

PROBABILITY EXPERIMENT BECAUSE

IT IS NOT REALLY CLEAR WHICH

OF THESE IS THE MOST LIKELY.

NOW, I COULD GIVE YOU AN

ARGUMENT, BUT I DON'T THINK

THAT THAT'S WORTHWHILE.

WHAT I THINK YOU NEED TO DO

IS TO ACTUALLY TAKE THE TIME

TO LIST IT OUT.

THE REASON I ASKED YOU TO DO

IT AS AN EXPERIMENT IS SIMPLY

TO GET THE SENSE OF WHAT IS

PROBABLY THE MOST LIKELY,

BECAUSE EXPERIMENTS GIVE YOU

PRETTY GOOD RESULTS IF THEY'RE

DONE CAREFULLY, BUT THEY'RE

NOT ALWAYS PERFECT SO YOU HAVE

TO BE CAREFUL ABOUT THAT.

SO PROBABILITY IS ONE OF THOSE

TOPICS OF MATHEMATICS WHICH I

ACTUALLY REALLY LOVE, BUT IT'S

SOMETIMES REALLY HARD TO GET

YOUR MIND AROUND EXACTLY

WHAT SHOULD HAPPEN BASED

ON EXPERIMENTAL RESULTS.

NOW, I'M ONLY GOING TO TAKE A

MOMENT OR TWO TO CLOSE OFF THE

CLASS BECAUSE WE DON'T

HAVE A WHOLE LOT OF TIME.

A blue slate appears that reads “Counting Problems, Combinatorics and Potpourri of Challenges. Complete assignments numbers 10 and 11 this last week.”

Mister C. continues I WILL LET YOU KNOW THAT THE

LAST PROGRAM OF THIS SERIES

WILL BE NEXT WEEK.

THE PROBLEMS ARE FROM

EXERCISES 10 AND 11.

EXERCISE 10 IS ABOUT COUNTING

PROBLEMS AND COMBINATORICS.

NOW, YOU DON'T HAVE TO KNOW

EXACTLY WHAT COMBINATORICS IS

TO DO THIS

PARTICULAR EXERCISE.

YOU JUST HAVE TO LOOK AT THE

QUESTION AND TRY TO FIGURE OUT

HOW YOU WOULD COUNT

THE NUMBER OF WAYS

THAT SOMETHING WILL OCCUR.

I WOULD HIGHLY RECOMMEND THAT

YOU WORK WITH A PARTNER

ON THAT PARTICULAR ONE.

THE OTHER ONE IS, I

CALL IT A POTPOURRI.

POTPOURRI MEANS JUST A VARIETY

OF QUESTIONS THAT COME FROM A

WHOLE VARIETY OF DIFFERENT

LITTLE BITS OF MATH.

SOME OF IT'S KIND OF

GEOMETRY-LIKE, SOME OF IT'S

NUMBER SENSE-LIKE, SOME OF

IT'S PROBABILITY-LIKE, AND

IT'S LIKE HAVING A BUNCH OF

LITTLE PUZZLES, LIKE PERHAPS I

STARTED AT THE BEGINNING OF

MY PROGRAM, BUT THEY'RE ALL

PUT TOGETHER.

SO I WOULD LOVE FOR YOU TO

TRY AS MANY OF THOSE PROBLEMS

AS YOU POSSIBLY CAN.

AGAIN, I WOULD HIGHLY

RECOMMEND THAT, IN FACT,

YOU WORK WITH A

PARTNER OR SOMETHING.

NOW, I HAVE A PHONE CALL?

A voice says YES.

Mister C. continues AND IT'S EITHER

RYAN OR FRANCIS.

The voice says I JUST WANTED TO FIND OUT WHY

OUR SCHOOL IS THE ONE BEING

CALLED ALL THE TIME.

Mister C. continues IT'S BECAUSE THE OTHER

SCHOOL HAS NOT GOT A VIDEO

CONNECTION TODAY.

The voice says OH.

Mister C. continues AND, THEREFORE, THEY CAN'T

SEE ANYTHING I'M DOING,

AND I'M NOT REALLY

TRYING TO PICK ON YOU.

I WOULD HAVE CERTAINLY TALKED

TO BOTH SCHOOLS AS MUCH AS I

COULD HAVE, BUT I GUESS

IT'S ONE OF THOSE THINGS

WITH THE BAD WEATHER.

The voice says OKAY.

Mister C. continues OKAY?

NOW, SPEAKING OF QUESTIONS

AND ANSWERS, DO YOU HAVE ANY

QUESTIONS BEFORE

I SIGN OFF TODAY?

AND I'LL GIVE YOU

A MOMENT OR TWO.

WELL, I'M SEEING NO QUESTIONS.

I'M LOOKING FORWARD TO

SEEING YOU ON THURSDAY

OF NEXT WEEK FOR OUR FINAL

PROGRAM, AND WE'LL SEE YOU THEN.

SO LONG.

BYE-BYE.

A green slate appears on screen. It shows a text that reads “Please remember to log off! Pick up handset. Press number sign then seven. Press 1 to confirm. Hang up handset. See you next time!”