We received lots of excellent solutions to this problem. Well done to students from Raines School, Lancaster Lane School, Risley Primary School, Garden International School, Glan-Y-Mor, Selside School, Kings School, Beijing City International School, Cirencester Kingshill School, Highfields School, Winster C of E School and Balfour Junior School for all finding solutions which give numbers greater than 25.

Alison managed to find numbers 1 to 46 with this daisy: 1 in centre, 2,4,12,19,8 around the edge.

Ria's method for making a daisy with numbers 1 to 28 can be found here.
 
Simon and Will made a daisy with numbers 1 to 37 and explain their method below:

To begin, we used the same flower as NRICH. We put 1 in the middle because we decided that it would be the most used number. We put 2 in next to make 2 and 3. Then we put a 4 in and we could make numbers 5,6,7. We wrote the numbers we could make underneath the flower. We then added 8 to make 8,9,10. We put 8 next to the 2 and we then could make numbers up to 15. We then added 12 and put it next to the 4 to make 16. The 10 was the last number to be added. We added 8 + 10 + 12 + 4 + 2 + 1 to make the biggest number 37. We then tried to make all the numbers in between and it worked.
  
Here are some more solutions.
 
Emma's daisy makes all the numbers 1 to 38:


Tamsin's daisy makes all the numbers from 1 to 40:

 
 
Carolyn's daisy makes all the numbers from 1 to 43:
 
 

Stephen's daisy makes all the numbers from 1 to 45:


Students from Riccarton High School found another daisy that makes all the numbers from 1 to 45:
 
They checked this worked by entering the six numbers in the cells of a spreadsheet and reproduced the numbers in six columns with the total on the right. It was then necessary to go through deleting some numbers on each line to give the totals 1 to 45 as in Ali's solution below:
 

1	21	8	9	2	4	Total
1						1
				2		2
1				2		3
					4	4
1					4	5
				2	4	6
1				2	4	7
		8				8
1		8				9
			9			10
			9	2		11
1			9	2		12
1		8			4	13
1			9		4	14
			9	2	4	15
1			9	2	4	16
		8	9			17
1		8	9			18
		8	9	2		19
1		8	9	2		20
	21					21
1	21					22
	21			2		23
1	21			2		24
	21				4	25
1	21				4	26
	21			2	4	27
1	21			2	4	28
	21	8				29
	21		9			30
1	21		9			31
	21		9	2		32
1	21		9	2		33
1	21	8			4	34
1	21		9		4	35
	21		9	2	4	36
1	21		9	2	4	37
	21	8	9			38
1	21	8	9			39
	21	8	9	2		40
1	21	8	9	2		41
	21	8	9		4	42
1	21	8	9		4	43
	21	8	9	2	4	44
1	21	8	9	2	4	45