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Hard Brain Teasers

Genuinely challenging mathematical problems requiring careful, multi-step reasoning.

Easy (15) Intermediate (10) Hard (4) ← Back to Quantnotes

4 puzzles recovered from the original Quantnotes.com Edutainment section (c. 2001–2008). Click a title to read the puzzle, then press Show Answer to reveal the solution.

01 The Five Bags of Coins ▾

You are given 5 bags containing 100 coins each. The bags can contain
coins of 3 different types that look identical. The first type weighs
9 grams, the second type 10 and the third type 11 grams. Each bag contains
coins of equal weight but you do not know how many of the 5 bags are of
the different types. (i.e. all 5 bags might well contain 9 gram coins
as far as you are concerned). You are given a huge digital balance. How
many times do you need to use the balance to clearly dertermine the type
of coin contained in each bag?
Hint : Do the Copper/Gold
coin problem first.

Show Answer ▾

It can be done in one measurment only. The way to reason
here is to first label the bags 1 to 5 then extract a certain number of
coins from each bag in such a way that the total sum of the weights of
the coins extracted corresponds to one and only one possible configuration
or arrangement of the weights for each bag. If there were just 2 types
of allowed coins then the number to extract from each bag would be 16
...(i.e. powers of 2). In our case with three types of coins we need to
extract 81 coins...(i.e. powers of 3). No wonder each bag has
to contain more than 81 coins ... 100 is a good choice!

02 Ball Balance ▾

You have 12 identical ball one of which is heavier OR lighter than and
the rest (you don't know which). Using just a balance and 3 measurements,
how can you determine which is the defective ball?
Hint : first do it for 9 identical balls and using only 2 measurements
knowing that one is heavier (or lighter) than the rest.

Show Answer ▾

The solution to this one is very lengthy to write down.
It starts by dividing the balls into 3 groups of 4 and comparing any two
groups together. There are two possible outcomes: they balance, so the
odd ball is the third group of 4 ball; or, they don't balance. Therefore,
the odd ball is in one of the 8 on the balance (harder), but at least
we have a identified four balls we know are non defective to determine
if the ball is heavier or lighter in the next two measurements. Extra
information is needed to be gather by noting any changes in direction
of the balance when moving balls about for subsequent measurements to
solve this problem.

03 Couple Permutation Problem ▾

If the girlfriends of three couples are seated in fixed positions around
a table, the is only one possible way to seat their boyfriends so that
men and women alternate and no boyfriend is next to his girlfriend. With
four couples, there are two possible arrangements.
How many possible arrangements are there with five, six and ten couples?

Show Answer ▾

13, 80 and 439,792 respectively.
Lots of people wrote to ask us how we get to the answers,
well, perhaps this might help:
http://mathworld.wolfram.com/MarriedCouplesProblem.html

04 Einstein's Test ▾

Einstein wrote this test and said that 98% of the people can't solve
it... Let's hope you are one of the remaining 2%.
There's only one solution for this problem.
There are 5 houses of 5 different colours.
In each house lives a person of a different nationality.
Those 5 people drink different drinks, smoke cigarettes of a
different brand and have a different pet.
None of them has the same pet, smokes the same cigarette or drinks
the same drink.
Question: Who has the fish?
Tips:
The englishman lives in the red house.
The swedish has a dog as a pet.
The Dane drinks tea.
The green house is at the left side of the white one.
The person who lives at the green house drinks coffee.
The person who smokes Pall Mall raises birds.
The owner of the yellow house smokes Dunhill.
The man that lives in the house that is in the middle drinks
milk.
The norwegian lives in the first house.
The man that smokes Blends lives next to the one that has cats.
The man that raises horses lives next to the one that smokes
Dunhill.
The man that smokes Bluemaster drinks beer.
The german smokes Prince.
The norwegian lives next to the blue house.
The man that smokes Blends is neighbour of the one that drinks water.
Do give it a go before looking at the answer...

Show Answer ▾

Each national lives in a house, drinks, smokes and has a pet:
Nationalities: English, Swedish, Dane, German and Norwegian
Houses: Red, Yellow, Green, White and Blue
House Positions: 1st, 2nd, 3rd, 4th and 5th
Drinks: Beer, Tea, Water, Milk and Coffee
Pets: Horses, Birds, Cats, Dog and Fish
Smokes: Pall Mall, Bluemaster, Dunhill, Prince and Blends.
So the correct groupings should be:
English: Red House, 3rd, Milk, Birds, Pall Mall
Swedish: White House, 5th, Beer, Dog, Bluemaster
Dane: Blue House, 2nd, Tea, Horses, Blends
German: Green House, 4th, Coffee, Fish, Prince
Norwegian: Yellow House, 1st, Water, Cats, Dunhill