### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Tea Cups

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

### Exploring Wild & Wonderful Number Patterns

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

# Sending and Receiving Cards

##### Stage: 2 Challenge Level:

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 well-presented 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 2-5 6 children: 30 cards 7 children: 42 cards 8 children: 56 cards 9 children: 72 cards The first digits go from 7-15, 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.

3 x 4 = 12 . . . 12 cards sent altogether.

5 x 4 = 20 . . . 20 cards sent altogether.

8 x 7 = 56 . . . 56 cards sent altogether.

15 x 14 = 210 . . . 210 cards sent altogether.

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;