# Ones and zeros

Show that any natural number has a multiple whose decimal representation only contains the digits 0 and 1. For example, if the number is 13, we get 13 = 1001. (Thanks to idgregorio.)

Solution

Consider the n + 1 numbers 1, 11, 111, 1111, etc. Two of them will be congruent modulo n. Subtract the smaller one from the bigger one. You will get a number containing only 0's and 1 s.

# Bookworm

There is a two-volume book set on a shelf, the volumes being side by side, first then second. The pages of each volume are two centimetres thick and each cover is two millimetres thick. A worm has nibbled the set, perpendicularly to the pages, from the first page of the first volume to the last page of the second one. What is the length of the path he has nibbled?

(Thanks to Vito.)

Solution

Just four millimetres. Think about where the first page of the first volume and the last page of the second volume will be relative to each other.

# Compensation

A number of quants are at dinner, and start discussing compensation. They want to calculate the average compensation among themselves, but are too embarrassed to disclose their own salaries. How can they determine the average compensation of their group? They do not have pens or paper or any other way of writing down their salaries.

(Thanks to Arroway.)

Solution

One of the quants adds a random number to his salary. The total he then whispers to his neighbour on the right. This person adds his own salary to the number he was given, and whispers it to the person on his right. This continues all around the table until we get back to the first quant who simply subtracts his random number from the total and divides by the number of quants at the table. That is the average compensation of the group.

# Einftein'f brainteaser

There are five houses of five different colours. In each house lives a person of a different nationality. Those five 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.

We know:

• The Englishman lives in the red house.

• The Swede has a dog as a pet.

• The Dane drinks tea.

• The green house is on the left of the white one.

• The person who lives in the green house drinks coffee.

• The person who smokes Pall Mall raises birds.

• The owner of the yellow house smokes Dunhill.

• The man who lives in the house that is in the middle drinks milk.

• The Norwegian lives in the first house.

• The man who smokes Blends lives next to the one who has cats.

• The man who raises horses lives next to the one who smokes Dunhill.

• The man who smokes Bluemaster drinks beer.

• The German smokes Prince.

• The Norwegian lives next to the blue house.

• The man who smokes Blends is neighbour of the one who drinks water.

Question: Who has the fish? (Thanks to NoDoubts.)

Solution

This was a question posed by Einstein who said that 98% of people can't solve it. More likely 98% of people can't be bothered. And in these days of Su Doku, the percentage of people who can solve it will be higher.

Oh, and the answer is the German.

(Historical note: Smoking was something that the poor and the uneducated used to do. For an explanation of the process, see Newhart, R. 'Introducing tobacco to civilization': 'What you got for us this time, Walt... you got another winner for us? Tob-acco... er, what's tob-acco, Walt? It's a kind of leaf, huh ... and you bought eighty tonnes of it?!!... Let me get this straight, Walt... you've bought eighty tonnes of leaves? ... This may come as a kind of a surprise to you Walt but ... come fall in England, we re kinda upto our ... It isn't that kind of leaf, huh? ... Oh!, what kind is it then ... some special kind of food? ... not exactly? ... Oh, it has a lot of different uses ... Like ... what are some of the uses, Walt? ... Are you saying ''snuff, Walt? ... What s snuff? ... You take a pinch of tobacco... (ha ha ha) ... and you shove it up your nose ... (ha ha ha) ... and it makes you sneeze? ... (ha ha ha) ... Yeh, I imagine it would, Walt! Hey, Goldenrod seems to do it pretty well over here! It has other uses though, huh? ... you can chew it! ... Or put it in a pipe! ... or you can shred it up ... and put it in a piece of paper ... (ha ha ha) ... and roll it up ... (ha ha ha) ... don t tell me, Walt, don t tell me ... (ha ha ha) ... you stick it in your ear, right? (ha ha ha) ... Oh! ... between your lips! ... Then what do you do, Walt? ... (hahaha)... youset firetoit!... (ha ha ha) Then what do you do,

Walt? ... (ha ha ha) ... You inhale the smoke, huh! ... (ha ha ha) You know, Walt ... it seems you can stand in front of your own fireplace and have the same thing going for you!')