We have had an overwhelming reponse to this Penta Problem. Thank you to everyone who sent in a solution.

Particularly well explained answers came from Tilly at Maldon Court Preparatory School, Alistair from Histon and Impington Infant School, Caroline from Tattingstone School, Samuel from Bispham Drive Junior School, Miss White's Y3/4 Class, Lucy from The King's Primary School and Saloni who didn't give an age or school.

Let's look at the spells in turn:

1. 1, 2, 4, 8, 16, 32.
"We are doubling the number before it," Alistair explained.

2. 6, 13, 20, 27, 34, 41
Tilly wrote here that we are adding 7 to the previous number.
Samuel adds, "I worked out my 7 times table to help me because all the numbers are one less than multiples in the 7's table."

3. 127, 63, 31, 15, 7, 3
There were several different ways to work this one out:
" We are halving and taking away 0.5 from the answer," said Alistair. Lucy also tackled this one in the same way.
"The differences were halving themselves this time. The difference between 127 and 63 is 64; the difference between 63 and 31 is 32; the difference between 31 and 15 is 16 so I knew that the next differences would be 8 then 4," explained Samuel. Saloni also tackled it this way.
Miss White's class suggested that you -1 then divide by 2. David put this slightly differently by saying, "-1 x 0.5."

4. 1, 3, 6, 10, 15, 21, 28
Samuel suggests: "This time the difference increased by one each time. The difference between 1 and 3 is 2; between 3 and 6 is 3; between 6 and 10 is 4; between 10 and 15 is 5 so I knew that the next differences would be 6 and 7."
Miss White's class put it slightly differently, "Between the numbers there is a pattern of addition. You first add 2, then 3, then 4 and so on.
Saloni spots that these are all triangular numbers.

5. 2, 3, 6, 11, 18, 27, 38
David points out the pattern is +1 +3 +5 +7 +9 +11 +13 etc. Miss White's class pointed out that the numbers added are the odd numbers in order.

6. 1,8,27,64, 125, 216
Alistair says, "I spotted that several of the numbers were multiples, so I started by looking for factors before I spotted the answer. The nth term in the series would be nxnxn (or n cubed)."
Samuel looked at the difference between the numbers: This was the hardest one to solve. The numbers increased by 7, 19, 37 but I could not see a pattern so I looked at a pattern with 7, 19, 37. I saw that the differences between these numbers were 12, and 18, and were increasing by 6 each time. So I added 6 onto 18 which gave 24 then added 24 on to 37 which gives 61. Next I took 61 and added it to the 64 which gave 125. With the 24 I again added 6 which gave 30 so I added 30 to 61 which gave 91. I added 91 to 125 which gave a final number of 216."
Caroline, like Alistair, realised that all the numbers were cubes. "I just did 5x5x5 and 6x6x6 to get the answers for the next two numbers," she said. Lucy also knew these were cubic numbers.

7. 216, 168, 126, 90, 60, 36, 18, 6
As many of you found, this was hard to explain. Samuel has done an excellent job:
What I did for the final spell was I found the difference between 216 and 168 which was 48, between 168 and 126 which was 42, between 126 and 90 which was 36. I noticed that these differences decreased by 6 so I took 6 from 24 (60 - 36) which gave me 18. Then I took 18 from 36 which is 18 and so this is the next number in the sequence. Similarly, I took 6 from 18 (the last difference) which gave me 12 and then I took 12 from 18 (the 18 in the sequence) so the final answer is 6.
Alistair had a slightly different method: I found the factors of the numbers, and spotted that all are divisible by 6. When I had divided by 6, the series was 36, 28, 21, 15, 10, 6 and the differences went down by one each time.

Very well done! A brief mention also to everyone who sent in correct answers -

Everyone in the two Denbighshire masterclasses I worked with in March
Aimee from Emerson Valley
Zebedee, Rachael, Phoebe, Kirsty, Angelique, Andrew, Hugo, Tom, Lockyerm, Emma, Benny, Philip, Malcom and Thomas from Lioncubs, Wesley College Preparatory School in Australia
Milli from St Hilda's
Gareth aged 8 who lives in Frankfurt am Main
Aristo from Mater Christi Chatolic Primary School, Australia

But remember you must tell us how you did it!