### 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.

### Series Sums

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

# 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