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- In the US we often write dates like April 20, 2017, in the form 4-20-17.
In the Netherlands one prefers to write this date in the form 20-4-17.
If you don't know which one of the notations is used, how many dates in the
double-dash notation are ambiguous, each year?
- From a regular deck of 52 cards someone takes 3 cards and puts them face
down in a row on the table in front of us.
We are asked to name the 3 cards.
Luckily we have 4 spies.
The first spy whispers in our ear that he saw a king, and to the right of this
king is a queen.
The second spy whispers in our ear that she saw a queen, and to the left of this
queen is another queen.
The third spy whispers in our ear that he saw a heart, and to the left of this
heart is a spade.
The fourth spy whispers in our ear that she saw a spade, and to the right of
this spade is another spade.
Name the 3 cards.
- Two women mathematicians are sitting at a roadside cafe, talking about
family.
One mentions that she has 3 daughters the product of whose ages equals 36.
Remarkably, she mentions, the sum of their ages equals the number on the house
across the street.
The other replies that this is not enough information to find the ages of the 3
daughters.
True, says the one, but note that my oldest daughter has beautiful eyes.
Ah, says the other, but then I know your daughter's ages.
What are the ages of the 3 daughters?
- We want to time the boiling of an egg for 6 minutes.
All we have is a 4-minute hourglass and a 5-minute hourglass.
What is the quickest way to measure the 6 minutes?
- Click on this link to see this puzzle, the
tri-partition of a square.
- The Tribune paper has 16 pages on 4 double sheets.
One of these sheets is taken out, and the 4 page numbers are added together.
What is the sum?
A Sunday newspaper has 36 pages on 9 double sheets.
The same question:
When we take a sheet out and add the 4 page numbers, then what is the sum?
Explain your answers.
- A chess board has 8 by 8 equals 64 squares.
We are given 32 dominoes, each the size of 2 adjacent squares.
- Explain why we can cover the chess board with these 32 dominoes.
- Mary has a little saw with which she cuts the top left and the bottom
right little squares off the chess board.
Before running away, she also takes one of the dominoes.
Question:
Can we cover the mutilated chess board with the remaining 31 dominoes?
- Put the digits 1 through 8 in the eight green squares below such that
no consecutive digits occur in boxes that touch by an edge or that touch by a
vertex.
- Jack is twice as old as Jill was, when Jack was as old as Jill is now.
Together Jack and Jill are 28 years old.
How old is Jack?
How old is Jill?
- We are given a 7-minute hourglass and an 11-minute hourglass.
With these, what is the quickest way to time the boiling of an egg for 15
minutes?
- At the office's recent potluck I noticed that all except two of the
dishes were salads, all except two were pastas, and all except two were
casseroles.
How many of each dish were at the potluck?
- Getting ready for MATH 1300, his favorite class, Joe still needs to
put on socks.
In his sock drawer there are 12 pairs of white socks, 6 pairs of black socks,
and 3 pairs of red socks.
The only problem is they are all mixed up randomly throughout the drawer.
How many socks does Joe need to take (without looking) before he can be sure
to have at least one matching pair?
- Click on this link to see this puzzle, a
triangle of dimes.
- Suppose we have a cube of cheese.
This cube is divided into 27 small cubes of cheese, so it looks like Rubik's
cube (3 x 3 x 3).
There is this little mouse who wants to eat the whole cube of cheese.
She starts eating the cube in the center, and then has to continue eating
adjacent cubes, where two cubes are called adjacent if they share a face.
Can the mouse eat the whole cube of cheese?
- Jared ask his grandfather Danny how old he is.
Being a clever mathematician, Danny answers:
"All I am telling you is the following:
I have 6 children, and there are 3 years between each one and the next.
I had my first child when I was 21.
Now the youngest is 21."
Question:
How old is Danny?
- 6829, a prime number, can be expressed as the sum of two primes in
exactly one way.
Question:
What is the larger of the two primes whose sum is 6829?
- Click on this link to see this puzzle,
matches and 3 squares.
- William, a logician, and Sam, a computational scientist among other
things, spin a coin on the bar to see who pays for each round of root beer.
After a couple of hours William observes that he has won way more times than
Sam, despite the fifty-fifty chances with the coin.
So William offers the following deal.
Sam gets 2 fair coins and William gets 1.
Each time Sam tosses more heads than William, Sam wins.
What is the chance for Sam to win under this novel arrangement?
- A woman either always answers truthfully, always answers falsely, or
alternates true and false answers.
How, in two questions, each answered by yes or no, can you determine whether
she is a truth-teller, a liar, or an alternator?
- "Our math teacher has more than a million good ideas," says Alli.
"He does not," says Emily, "he has fewer good ideas."
"He has at least one good idea," says Brigid.
If only one of the 3 statements is true, how many good ideas does our math
teacher have?
- Click on this link to see this puzzle, the fish.
- On an island in the Pacific there live 850 women.
Four percent of them wear one earring.
Of the remaining 96 percent half wear no earrings and the other half wears two
earrings.
How many earrings do the women together wear?
- Hunter Jack wants to shoot a bear.
He leaves his camp, and walks 10 miles south where he perceives bear tracks.
Jack follows the bear tracks for 10 miles east, where he sees the bear and
takes a shot at it.
He misses.
Then Jack walks 10 miles north and is back at camp.
What is the color of the bear?
- The collection of four integers 1, 3, 8, and 120 is such that the
product of any two of them is 1 less than a perfect square.
Find a fifth number so that this property still holds when this fifth number is
added to the collection.
- A woman travels 2500 kilometers by car, with one spare tire.
Occasionally she rotates tires so that each tire is used an equal number of
kilometers.
For how many kilometers is each of the tires in use?
- Melania and Donald play a game with a pile of 100 jelly beans, where
they alternate in taking 1, 2, or 3 jelly beans from the pile.
Winner is the one who takes the last jelly bean.
Donald begins.
For which of the two is there a winning strategy, and what is that strategy?
- Five couples meet at a dance competition for pairs.
Michelle and Barry are one of the couples.
All shake hands with some of the members of the other couples.
No-one shakes hands twice with the same person.
Barry asks all other people including Michelle with how many people they shook
hands.
Surprisingly they all gave a different answer.
Question:
With how many people did Michelle shake hands?
- Click on this link to see 8 figures.
Complete the last row to continue the pattern of the 1st and 2nd row.
- These five problems are in this odt
file.
- This puzzle has four unrelated parts, all in
this file.
- We want to time the boiling of an egg for 6 minutes.
All we have is a 4-minute hourglass and a 5-minute hourglass.
What is the quickest way to measure the 6 minutes?
- Let us count on the fingers of our left hand as follows.
Our little finger is 1, our ring finger is 2, our middle finger is 3, our
index finger is 4, and our thumb is 5.
Keep counting by moving in the opposite direction:
The index finger is 6, the middle finger is 7, the ring finger is 8, and the
little finger is 9.
Keep counting by again reversing direction;
The ring finger is 10, the middle finger is 11, and so on, each time reversing
direction.
Which finger do we end on when we count to 2017?
- Click on this link to see this puzzle,
matches and 5 squares.
- We are given 12 identical matches.
We can arrange them as edges of polygons in many ways, like:
- When we line them up as boundaries of a square with sides 3, then the
enclosed area equals 9.
- When we line them up as boundaries of a rectangle of width 5 and
height 1, then the enclosed area equals 5.
- If we arrange them as boundary of the blue area below, then the
enclosed area equals 5.
Find a polygon arrangement such that the enclosed area equals 6.
(There is one beautiful answer.)
- A student on her way to Rome reaches a fork in the road.
One way continues to Rome, and the other is a dead end.
Luckily there sit on a bench two brothers, named Ivan and Jose.
Unfortunately one of them always lies.
Fortunately one of them always tells the truth.
She knows not which one is the truth teller, and which one is not.
What one question should she ask to one of the people to get the correct answer
for her quest for the way to Rome?
- On the kitchen table lies a row of 30 coins, a mixture of pennies,
nickels, dimes, and quarters.
Alternatingly Elizabeth and Jane take one coin from the table, but with the
condition that each can only take a coin from either end of the row.
How can beginner Elizabeth guarantee that she ends up with at least half of the
total value of the coins?
- Find the sum 1+2+3+ ... +199+200, and show through your work why the
answer is correct.
(Clever mathematical methods are encouraged.)
- In the dark of the night four people have to cross a rickety suspension
bridge over a deep ravine.
The bridge carries at most 2 people at a time, and the group has only one
lantern.
It takes Kristen 1 minute to cross the bridge.
It takes Anna 2 minutes to cross the bridge.
It takes Alyssa 5 minutes to cross the bridge.
It takes William 10 minutes to cross the bridge.
How can all cross the bridge in the least amount of time?
- This puzzle consists of 3 parts, each next one slightly more work.
- You are given a balance and 3 identical looking coins.
2 of the 3 coins are pure gold.
The 3rd coin is not pure gold, and is slightly lighter than a proper gold coin.
Show how to use the balance exactly 1 time to determine which of the 3 coins is
the one that is slightly lighter.
No trickeries like putting coins on the scales one at the time, just 1 weighing
allowed.
- You are given a balance and 9 identical looking coins.
8 of the 9 coins are pure gold.
The 9th coin is not pure gold, and is slightly lighter than a proper gold coin.
Show how to use the balance exactly 2 times to determine which of the 9 coins
is the one that is slightly lighter.
No trickeries like putting coins on the scales one at the time, just 2
weighings allowed.
- You are given a balance and 27 identical looking coins.
26 of the 27 coins are pure gold.
The 27th coin is not pure gold, and is slightly lighter than a proper gold
coin.
Show how to use the balance exactly 3 times to determine which of the 27 coins
is the one that is slightly lighter.
No trickeries like putting coins on the scales one at the time, just 3
weighings allowed.
- Click on this link to see 5 figures.
One of these 5 figures is most different from the other 4.
Which one is most different, and why?
- Three cannibals and three missionaries (probably Jesuits) want to cross
the river.
The boat holds at most two people.
The problem is that if on either side at any time the cannibals outnumber the
missionaries, those missionaries will be eaten.
How can all six get to the other side alive?
- You have bags A, B, and C, each containing two marbles.
Bag A contains two red marbles, Bag B contains two blue marbles, and Bag C
contains one red marble and one blue marble.
You pick at random a bag and one marble out of that bag.
It is a red marble. What is the probability that the remaining marble from the
same bag is also red?
- At a restaurant, how could you choose one out of three desserts with
equal probability with the help of just one fair coin?
- Both Al and Ben sell a basket of apples on the market.
They have the same number of apples, but Al's are Honey Crisps and Ben's are
Braeburns.
So Al can sell his at 2 for 1 dollar, and Ben can sell his at 3 for 1 dollar.
Friend Chris walks by and is asked to sell the apples of Al and of Ben, who
both have to go to a nearby bar to talk busines.
In their absence, Chris simplifies his job by mixing the apples of Al and Ben
in one large basket, and sell them at 5 for 2 dollars.
By the end of the day Chris is 1 dollar short.
How many apples did Al and Ben originally have?
- Eight thousand seven hundred and six is written as the number 8706.
How do we write fifteen thousand fourteen hunderd and thirteen?
- Eight thousand eight hundred and eight dollars can be properly written
as $ 8808.
How can nineteen thousand nineteen hundred and nineteen dollars be written
down in such a proper way?
- Why did the chicken cross the Möbius strip?
- In the millenial year 2000, an oldtimer tells his grandchildren that he
was n years old in the year n^2 (n squared).
In what year was the oldtimer born?
- We stack 1000 identical size tiny cubes in the straightforward way to
form one big single cube of 10 by 10 by 10.
How many tiny cubes share a corner of the big one?
How many tiny cubes share an edge of the big one?
How many tiny cubes share a surface with the big one?
How many tiny cubes are completely inside the big one?
- In a game of bridge a deck of 52 cards is (randomly) dealt to 4
players around a card table.
Each player gets 13 cards.
Players sitting opposite one another form a team.
So there are 2 teams of 2 players each.
What is more likely, that our team holds all hearts, or that the opposing team
holds all hearts?
- Recall that a chess board has a size of 8 by 8 squares.
- First, try to cover the chess board with dominoes, where each domino
has a size exactly 2 squares in a row (of course you can.).
How many dominoes do you need?
- Next, try to cover the original chess board with trominoes, where each
tromino has exactly the size of 3 squares in a row.
If it is impossible, explain why it is impossible.
- Finally, try to cover the chess board with trominoes such that exactly
one field is left uncovered.
How many trominoes do you need?
- This one should be very easy.
Find a statement about numbers n that is true exactly when the value of n is
bigger than a hundred.
- 5 mathematicians from the Netherlands walk along the street.
At once they notice that someone had dropped 10 pennies across the street.
The 5 Dutchmen dash through traffic across the street to pick up these pennies.
All 5 of them end up with a different number of pennies.
How many pennies did each of them end up with, and why?
- This is a puzzle which occurs in one of the Simpsons episodes.
Click on this link to see 4 figures.
Add a 5th figure that continues the pattern.
- A ten-story student residence has identical stairs between its floors.
How many times does it take to climb from the first to the tenth floor as it
is to climb from the first to the fifth floor?
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Behind a screen I put three cards on a table, a King of hearts, a Queen of
hearts, and a Jack of hearts.
- If all faces are up or if all faces are down, I'll yell JACKPOT.
You are allowed to give me instructions to flip one particular card, for
example instruct me to flip the Queen.
Find a shortest list of such instructions to guarantee that I'll yell JACKPOT
at least once.
- The same question as above, except that I'll only yell JACKPOT when
all cards are face up.
- Luke, Stan, and Barry share a room in a cheap motel, to save money.
The three intend to equally share in the cost.
Luke collects 7 dollars from both Stan and Barry, and adds his own 7 dollars.
When Luke goes to the front desk in the main lobby and hands over the 21
dollars, the clerk observes that he is overpaid by 5.
So the clerk returns 5 singles to Luke.
On his way back to the room, dishonest Luke decides to pocket 2 dollars for
himself, but otherwise returns a single dollar to all.
So all have paid 6 dollars for a total of 18, while Luke keeps an extra 2
dollars for himself, for a total of 20 dollars.
Where is the missing dollar?
- Click on this link to see the queens puzzle.
- I write 5 individual letters for 5 people, and I write their 5 addresses
on 5 envelopes.
If I put the 5 letters randomly in the 5 envelopes, what is the chance that
exactly 4 of the letters are in the right envelopes?
- Find three positive integers whose sum equals their product.
(It is harder to show that it is essentially the only solution.)
- On a national homework problem for 729600 students, 3.5% scores 1 point.
Of the remaining 96.5% of the students, half get 2 points and half get 0.
How many points in total did the students score?
- Two related questions.
- Find the sum 100 + 99 + 98 + 97 + ..... + 4 + 3 + 2 + 1.
Show how you got the answer.
Clever mathematical methods are much recommended.
- Find a closed formula for the sum 1 + 11 + 21 + 31 + ..... + (10n-39)
+(10n-29) + (10n-19) + (10n-9) of the first n numbers with right-most digit 1.
Include your work showing why you formula is correct.
- A farmer wants to move a wolf, a goat, and a cabbage across the river.
His boat allows him to move only 1 item at a time.
If left alone, the goat eats the cabbage, and the wolf eats the goat.
How can the farmer get all across the river unharmed?
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An urn contains 1000 yellow marbles and 881 blue marbles.
We have a large pile of yellow marbles outside, more than enough.
We are asked to empty the urn by repeating the following process.
Randomly pick 2 marbles from the urn.
There are 3 possibilities.
(1) If both marbles are yellow, then put one of them back in the urn, and leave
the other out.
(2) If one marble is yellow and one is blue, then put the blue one back and
leave the yellow one out.
(3) If both marbles are blue, then leave them both out, but put a yellow one
back in the urn.
Finally there is 1 marble left in the urn.
What is the color of this last marble?
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This one dates from 19 November 2021 or earlier.
Whole numbers p and q are both not divisible by 2 or 3.
Show that p^2 - q^2 is divisible by 24.
- The princess has a thousand suitors.
She likes at least one of the suitors, but of each two suitors she dislikes at
least one.
How many suitors does she like, and how many suitors does she dislike?
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One glass has 5 ounces of pure water, the other glass has 5 ounces of pure
wine.
We pour a bit of the water into the wine glass, after which we pour a same
quantity of the diluted wine into the glass with water.
So both glasses again have 5 ounces.
Did we put more water in the wine, or did we put more wine in the water?
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