A problem to think about when you have nothing else to do
Once upon a time there were three bears and they all started eating
equal portions of porridge at the same time. Father bear ate his all up in
ten minutes. Then he joined in with Mother bear’s unfinished bowl,
which he and she took another five minutes to empty. Father and Mother
then joined in with Baby bear and all three took five minutes more to
finish.
How long would Baby bear have taken on his own?
Birthdays
What are the chances that within this room there are at least one pair
with the same birthday?
With 2 people the probability of the 2nd not having the same birthday as
the first is 364/365.
With 3 people the probability of the 3rd not having the same birthday, as
the other two is 363/365.
The probability of the 5th person not having the same birthday in common
with the other 4 is 364x363x362x361/365^5.
You take this from 1to get the probability that at least 2 have the same
birthday.
For a group of 23 the probability is 0.5 so for a group of 24 or more you
are more likely than not to have 2 people with the same birthday.
Try it!!
Frogs
Is an ageless classic that can be played with people, counters or
on the computer. The traditional starting point is 3 boys and 3
girls sitting on a line of 6 chairs with an extra chair between
them. A move is one person sliding into an empty chair or jumping
over one other person. Can the girls and the boys swap places?
How many moves are needed?
Frogs is available on the Internet at
http://www.hellam.net/maths2000/frogs.html
Fleas
Seat five people side by side in a straight line. A move is defined as
one person
changing from a sitting position to a standing
position, or from a standing position to a sitting position.
A person can only move when both of the following conditions are
satisfied.
1.The person on their immediate left is standing.
2. All the other people on their left are sitting
The person on the extreme left has no restrictions on their
movement.
Can all five people be standing up together?
If so, what is the fewest number of moves in which this can be
achieved?
What if the group changes size?
What if the initial state is changed?
1089
Write down a 3-digit number, say
Reverse the digits
Subtract
Reverse the digits
Add
Do you always get 1089?
742
247
495
594
1089
What happens if you start with 564?
Can you prove that you will always get 1089?
Are there similar results with 2 digits?
Are there similar results with 4 digits?
Extend to other bases.
Why is 6174 special?
Take a 4-digit number. Stage 1: rearrange it in descending
order, largest digit first. Stage 2: reverse it and stage 3:
subtract the smaller number. Keep on repeating stage 1 to
3 until it becomes 6174 the Kaprekar Constant. If you
apply the stages to 6174 you get 6174 again. It is a loop!
Dr D R Kaprekar was born in Dahann, India in 1905. D.R.
Kaprekar, who loved doing puzzles and calculations from a
very young age, discovered this strange property in of the
number in 1946. Try it with 1234, 7777 and 2007. If the
number becomes 3-digits then add a zero.
Is there a Kaprekar constant for 3 digits?
Try 835. After only one stage the number starts looping. Is 495
the Kaprekar Constant for 3-digit numbers? Try 972 and some of
your own. What happens if two of the digits are the same? Try
833, 272, 211 and some more of your own.
Are there any numbers that don’t become the Kaprekar
Constant?
What is a Niven Number?
Another curious thing about the number 6174 is that it is
also a Niven number. That’s a number that can be divided
by the sum of its digits without any remainder.
6 + 1 + 7 + 4 = 18 6174/18 = 343.
Which of these numbers IS NOT a Niven Number? 252, 150,
36, 236, and 444.
Niven numbers are also called Harshad numbers. They
were called Niven numbers by American mathematician
Robert E. Kennedy in honour of Ivan Niven who mentioned
them at a conference on number theory in 1977.
The Game of 31
The game of 31 is a game for pairs. Four sets of numbered cards (1-6)
are placed in four rows. Turns are taken to remove a card and subtract
its value from a start of 31. The winner is the person to reach zero with
their last card. Ace’s to 6’s from a set of playing cards make an easy
and pleasant set to play with but the array below can also be used.
After finding all the winning strategies have some fun by using “what if
not questions” e.g. what if not 31?
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
Bell ringing
4 bells are rung in order
ABCD
The middle 2 bells change
ACBD and are rung again
Now the end bells change
CADB
This continues, middle, ends, middle, ends until the change is
complete
Can you complete a change without moving position?
Palindromes
Choose any number, reverse it and add
216
612
828
828 is a palindromic number (the same backwards as forwards)
Try 154. 605 is not but if we repeat then 1111 is
Does this always happen? Are all palindromic number multiples of
11? Take some palindromic numbers. Can you always make them by
adding a number and its reverse? When you can, are there rules
to find the numbers?
DO THE MATH, HOMER – Simon Singh
The Observer Review Sunday 22 September 2013.
During an episode called “Marge and Homer Turn a Couple Play”(2006) a baseball screen displays 3
multiple options for the game’s attendance.
8,128
8,208
8,191
Although these 3 numbers appear to be random they are in fact example of 3 special types of
number;
a perfect number, a narcissistic number and a Mersenne prime.
8,128 is perfect because the sum of its divisors add up to the number itself.
6 is the smallest because 1, 2, 3 add to 6
The next is 28 because 1, 2, 4, 7, 14 and to 28.
The next is 496 and the 4th is 8,128.
Rene Descartes (17th C French mathematician and philosopher pointed out that perfect number like
perfect men are very rare.
8,208 is a narcissistic number because it contains 4 digits which if each is raised the 4th power and
added gives the number itself.
i.e. 8^4 + 2^4 + 0^4 + 8^4 = 8208
As 8208 can recreate itself from within itself this suggest a number that is in love with itself –
hence narcissistic.
Within an infinity of numbers far fewer than 100 narcissistic numbers have been found.
8,191 is prime because it has no divisors except 1 and itself.
It is labelled a Mersenne prime after another French 17C mathematician, Marin Mersenne, who
spotted that 8,191 was equal to 2^13 – 1. In general Mersenne primes fit the pattern 2^p – 1, where
p is any prime number.
Thanks