Seasonal Problems. A selection of short (and not so short) problems . Calendars. This year, December 1 st is on a Saturday, in which year does this happen next? How many times this century will December 1 st fall on a Saturday?. Calendars.
Can you find a formula for this?
At a New Year’s Eve party there are some adults who all shake hands with each other… except there are a small (unfriendly) group who all refuse to shake hands with each other, but will shake hands with everyone else.
There are 135 handshakes in total. How many people are at the party? How many are there in the ‘unfriendly’ group?
How many different answers can you find for this?
This year Santa has to deliver to all
the young people of the world, roughly
2 billion under 18’s in total.
Assuming, on average, that there are 3.5 young people per household, how long does he have for each household to get down the chimney, eat the mince pie, grab the carrot for the reindeer and fill the stockings with goodies? (Because of the earth’s rotation he probably has about 31 hours available).
…and how many households is that per second or per minute? (whichever seems appropriate)
A class of 25 pupils have an advent calendar. The first pupil decides to open all the windows on the calendar. The second pupil goes and closes all of the windows that are a multiple of 2. The third pupil changes all the multiples of 3 – if they are open then she closes them, if they are closed then she opens them. The fourth pupil changes all the multiples of 4, if they are open then he closes them, if they are closed then he opens them. The fifth pupil changes the multiples of 5 and so on until the 25th pupil changes the multiples of 25.
When the teacher arrives – which windows on the calendar are open?
Can you explain why?
Mrs Claus was delighted to find an unusual shaped Christmas cake in her local supermarket; a regular hexagonal one instead of the usual circular or square ones. She bought one immediately and took it home to show to Santa.
Santa also loved the new cake shape but pointed out one small problem… “There are 5 of us eating and we’ll need to share the cake equally, how are you going to cut it up to give 5 equal portions?”
(the same volume)
“Easy”, said Mrs Claus…
How did she do it?
Adults shaking hands.
This problem continues the triangle numbers theme, but with a less obvious method of solution.
One way to solve it is to consider the first 20 triangle numbers which are given on the next slide. This could be displayed to pupils, but it would spoil the opportunity for pupils to think if it is shown too early on.
To find an answer for the numbers of people at the party and in the ‘unfriendly’ group simply search for a pair of triangle numbers with a difference of 135.
There are 3 possible answers for this:
The first possible issue with this problem is knowing how to write a billion: 1 x 109
or 1 000 000 000
Rounding answers to different degrees of accuracy will give slightly different answers:
You might like to pose some ‘localised’ questions, (or ask members of the class to pose some) such as how long Santa will take to deliver to all the pupils in school, or the local village, town or city, or the UK.
This problem is adapted from a ‘prison door’ problem.
Each door is initially opened by child 1 and each is then subsequently opened or closed by any child who’s ‘number’ is a factor of it.
The cycle clearly alternates:
open close open close etc.
Any number which has an even number of factors will end up closed; any number with an odd number of factors will end up open.
Hence, calendar doors with square numbers end up as the only open ones.
For the problem given of dividing a hexagon into five equal sections,
Draw a segment from the centre to any marked point
Firstly divide each side into 5 equal lengths.
Find the centre
Count round the hexagon marking every sixth segment
Can you prove that the 5 segments are equal?
Consider each of the small triangles… what is the base and height of each?
Would this idea work for other polygons and/or number of required equal sections?
Dynamic geometry would be helpful to demonstrate that the sections have an equal area, but also to show that if equal angles are taken at the centre, this does not result in sections of equal area.