Sending and Receiving Cards
Problem
There are many festivals and celebrations when we send cards to each other. Some schools organise a postal service within the school and some of the pupils get the job of being postmen/women.
Groups of children send cards to each other sometimes and that's really good for them all; it's a good feeling getting the cards.
Suppose there are three children, Georgie, Jo and Chris, and these three are really good friends. They decide to send cards to each other.
Georgie sends two cards, one to Jo and one to Chris.
Jo sends two cards, one to Georgie and one to Chris.
Chris sends two cards, one to Jo and one to Georgie.
So these three friends send six cards altogether. Also they get six cards altogether!!!!
What if in another class in that school there is a group of four children who are best friends and at this time of the year each sends the others a card.
Raj sends $3$ cards, one to Bex, one to Jon and one to Loo.
Bex sends $3$ cards, one to Raj, one to Jon and one to Loo.
Jon sends $3$ cards....
Loo sends $3$ cards....
So a lot more cards are sent by the four children altogether.
This season's challenge is to explore the number of cards that are sent altogether when there are $5$ children, then $6$, then $7$, etc. , up to perhaps a class of $30$ children who all send cards to each other and you work out how many cards are sent altogether.
Once you have this set of numbers it might be good to write them down underneath one another (a bit like you may have done when you did the investigation called Exploring Wild & Wonderful Number Patterns ).
You could now, if you have not done it already, start looking at some of the things that pop up in this number pattern. I'm not sure what they all are of course, but I've seen a few patterns, so get searching for what YOU can find!
Then you'll be able to ask, "I wonder what would happen if I ...?'' about so many things.
Getting Started
How will keep track of the number of children you have tried?
Student Solutions
We received many good quality solutions this month. A lovely Christmas present for us! We haven't included all of them, but instead we've chosen a variety of solution types to show you. Thank you to Robert, Sarah and Anna for your correct and wellpresented solutions.
From The Juniors: The Christian School (Takeley)
Michael: "To find the number of cards you have to times the number of people by the number of cards they sent.''
Class: "The number of cards sent is one less than the number of people.''
Our Table:
Number of people 
Number of cards sent 
1  0 
2  2 
3  6 
4  12 
5  20 
6  30 
7  42 
8  56 
9  72 
10  90 
11  110 
12  132 
Frances: "They are all even numbers.''
Lawrie: "The units digit goes 0, 2, 6, 2, 0, 0, 2, 6, 2, 0''
Michael: "The units digits are symmetrical in the table''
Class: "Let's carry on the number pattern:''
0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650,...
Lawrie: "The units digit repeats the pattern every five numbers.0, 2, 6, 2, 0''
David  with help from the class and much amazement:
"Every time there is a six you add one more ten to the tens column.''
30 3+1=4
42 4+1=5
56 5+2=7 You've hit the six so you add one more ten
72 7+2=9
90 9+2=11
110 11+2=13
132 13+2=15
156 15+3=18 You've hit the six so you add one more ten
182 18+3=21
210 21+3=24 etc
It does carry on.
Class: "While we were working on last month's dice problem (a bit late) we saw the pattern of triangular numbers and were playing around with squares and doubling the triangular numbers in order to find some patterns etc when:''
Samuel: "All the numbers in the card problem are double the triangle numbers!''
NB Sam is our Mr Memory Man.
Thank you for such an enjoyable problem. My class think you make things look easy and it's your fault that they get so complicated. Apparently you work some kind of magic Bernard!
Siobhan of West Flegg Middle School sent the following solution:
"4 x 3 = 12 
5 x 4 = 20 
6 x 5 = 30 
7 x 6 = 42 
8 x 7 = 56 
9 x 8 = 72 
10 x 9 = 90 
11 x 10 = 110 
12 x 11 = 132 
13 x 12 = 156 
14 x 13 = 172 
15 x 14 = 210 
16 x 15 = 260 
17 x 16 = 272 
18 x 17 = 306 
19 x 18 = 342 
20 x 19 = 380 
21 x 20 = 440 
22 x 21 = 462 
23 x 22 = 506 
24 x 23 = 552 
25 x 24 = 600 
26 x 25 = 650 
27 x 26 = 702 
28 x 27 = 756 
29 x 28 = 812 
30 x 29 = 870 
I worked out all of this because I wanted to find out if there was any pattern in the last numbers. I found out that there was. The pattern is 2, 6, 2, 0, 0 and it carries on about 6 times. Anyway, I found out that the whole class would send to each other all together 870 cards. I worked out the sum by doing this:
30 
x29 
 
270 
600

 
870

=== 
But, what if two of the friends fell out with each other?
What if they changed to a new school?
What if nearly all the class started to hate each other?
What if one of the children sent a card to the teacher?
Would any of these ifs change the sequence of the numbers at the end?''
Natasha of West Felff Middle School says:
"Jenny sends 4 cards: 1 to Paula, 1 to Sarah, 1 to Roxy, 1 to
Hannah
Paula sends 4 cards: 1 to Jenny, 1 to Sarah, 1 to Roxy, 1 to
Hannah
Sarah sends 4 cards: 1 to Jenny, 1 to Paula, 1 to Roxy, 1 to
Hannah
Roxy sends 4 cards......
Hannah sends 4 cards....
Therefore with a group of 5 children, 20 cards are sent
altogether
Steph sends 5 cards: 1 to Emma, 1 to Laura, 1 to Gemma, 1 to
June, 1 to Becky
Emma sends 5 cards: 1 to Steph, 1 to Laura, 1 to Gemma, 1 to June,
1 to Becky
Laura sends 5 cards......
Gemma sends 5 cards......
June sends 5 cards.......
Becky sends 5 cards......
Therefore with a group of 6 children 30 cards are sent.
''
Natasha continued using this method, then followed the pattern all the way up to 30 children, as shown below:
"Here are the final numbers of what I have done above, but up to 30 children:
5 children: 20 cards  The first digits of these numbers go from 25 
6 children: 30 cards  
7 children: 42 cards  
8 children: 56 cards  
9 children: 72 cards  The first digits go from 715, so they are jumping in twos 
10 children: 90 cards  
11 children: 110 cards  
12 children: 132 cards  
13 children: 156 cards  
14 children: 182 cards  
15 children: 210 cards  
16 children: 240 cards  There are two 2s, three 3s, two4s, two 5s, two 6s, two 7s and two 8s 
17 children: 272 cards  
18 children: 306 cards  
19 children: 342 cards  
20 children: 380 cards  
21 children: 420 cards  
22 children: 462 cards  Every single number is even because they have all got an even number for the last digit 
23 children: 506 cards  
24 children: 552 cards  
25 children: 600 cards  
26 children: 650 cards  
27 children: 702 cards  
28 children: 756 cards  
29 children: 812 cards  
30 children: 870 cards'' 
This is part of another solution sent in by Rachel from West Flegg Middle School. It shows another way to solve the problem.
"The number of arrows round the persons name represents how many cards they have sent and received, which is always one number less that the group number.''
Chris used a grid diagram to discover more patterns.
I see a pattern.
On a grid you end up with a diagonal line.
The numbers when multiplied are all even in the totals.
There is a pattern in the last digits of the multiplied numbers.
The pattern is 2, 0, 0, 2, 6, 2, 0, 0, 2, 6...
There is also a pattern where you multiply the number of people (first number) sending the cards by the same number less one (second number) because you don't send a card to yourself.eg:
4  x  3  = 12 
1 ^{st} = no. of people 
2 ^{nd} 
Class of 30 children; 30 x 29 = 870 cards sent.''
Here are results in a spread sheet form;