### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### Natural Sum

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural numbers.

### Prime AP

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

# Charlie's Delightful Machine

##### Stage: 3 and 4 Challenge Level:
Hannah from Wallington High School for Girls suggested a possible starting point:
A strategy is to perhaps type in a number one by one so start with 1 then
2,3,4 etc and recording each light that gets turned on.

The Year 9 Puzzle Club at Huddersfield Grammar School worked out their machine's rules as follows:

We wrote down which lights lit up for numbers 1-45, then wrote down the sequences for each colour.

Yellow: 1, 11, 21, 31, 41...
Blue: 11, 22, 33, 44...
Green: 10, 21, 32, 43...
Red: 4, 10, 16, 22, 28...

Then we worked out the Nth term of each sequence.

Yellow: 10N+1
Blue: 11N
Green: 11N-1
Red: 6N-2

Dylan, from Landau Forte, told us his findings:

I found out that yellow started on 3 and I had to add 3 each time. All the numbers were a mixture of odd and even. Next, red started on 7 and I had to add 7 each time. The rule was all the numbers were a mixture of odd and even. Then blue started on 4 and I had to add 9 each time. The rule for this one was also that the numbers were odd and even. Finally green started on 3 and I had to add 4 each time. The rule for this was that all the numbers were odd.

Laurynne from Wallington pointed out that if the sequence has the rule $an+b$ with $a$ and $b$ both even, the terms will all be even, but if $a$ is even and $b$ is odd, the terms will all be odd.

Aswaath, from Garden International School, tested the machine in the same way, but gave us some extra interesting information about his machine's behaviour:

First, I experimented with the machine, and got these results:

 Green Blue Red Yellow 2 10 5 0 12 22 8 12 22 34 11 24 32 46 14 36 42 58 17 48 52 70 20 60 ... ... ... ...

From this I figured out the nth term of the pattern for each colour:
Green: 10n - 8
Blue: 12n - 2
Red: 3n + 2
Yellow: 12n - 12

From this I figured out all the numbers that made blue, yellow and green light up are even. The numbers for red are a mixture of odd and even as they increase by threes every time.

To take the the investigation further, I also recorded the results for various combinations of colours, in the table below:

 G B R Y GY GB GR BY BR RY GYB RGY 2 10 5 0 12 22 32 - - - - - 12 22 8 12 72 82 62 22 34 11 24 92 32 46 14 36

From this I figured out the nth term of the pattern for the colour combinations:

Green + yellow: 60n - 48
Green + blue: 60n - 38
Green + red: 30n + 2