Question

At a laundromat, there are 12 washing machines available for the people to use. On a
particular day, a total of 42 people came and did their laundry where each person uses exactly
one washing machine. Show that if no washing machine is used by more than 6 people, then
at least 5 washing machines are used by at least 3 people.

Answer 1

pigeonhole principle ?

Answer 2

yes but i confused. there is a hint : Consider when the maximum number of people uses 4 washing machines. Show that
that there is one more washing machine being used by at least 3 people.

Answer 3

pigeonhole = machine
pigeon = people
right ??

Answer 4

Yes. say the first four washing machines are used by maximum number of people : 4*6 = 24

(24 people used the first 4 washing machines.)
(24 pigeons in first 4 holes)

Answer 5

how many more washing machines (holes) are there ?
how many more people(pigeons) are there ?

Answer 6

8 washing machines and 18 people.
15 people use 5 washing machine, then 3 people left.
thats why 1 more machine use by at least 3 people.
yes or no ?

Answer 7

18 pigeons and 8 holes
by pigeonhole principle, one hole must contain at least 3 pigeons

Answer 8

18/8 = 2.25

so there will be few pigeons left after assigning 2 pigeons to each hole

Answer 9

that means "one" of the 18 washing machines is used by at least 3 ppl

Answer 10

okay.. thats mean each of 7 holes used by 2 pigeon , so 14 pigeons. 18-14 =4 pigeons left. 4 = at least 3 in 1 more pigeonhole.. right ?

Answer 11

thats right, but if you look at the division : 18/8
clearly thats more than 2, that means one of the hole must contain more than 2 pigeons

Answer 12

yes.. okay..thank you very much for your help.. :)

Answer 13

np :) do you get how to conclude
"at least 5 washing machines are used by at least 3 people."

Answer 14

i dont know .. can you explain it ?

Answer 15

we just showed that at least one of the last 8 washing machines is used by at least 3 ppl

right ?

Answer 16

im not sure. but if you say so..then its right. let me try with my friends first. if not okay i'll ask you later. :)

Answer 17

ok sure take your time :)

Answer 18

```
18 pigeons and 8 holes
by pigeonhole principle, one hole must contain at least 3 pigeons
```

this shows that one of the 18 washing machines must be used by at least 3 ppl.

Answer 19

okay.. i'll try it.. thanks.. :)

Answer 20

yw:)