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!”

Watch: Counting on an Answer