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Factors and Multiples Game

Age 7 to 16
Challenge Level

Many thanks to everyone who shared their solutions with our team. We were very impressed by your resilience and perseverence to produce the chains that you submitted to NRICH. Several of you sent in more than one solution, substituting an earlier chain with an even longer one after spending more time on the challenge. Well done to you all!

We've published below a selection of increasingly long chains which have a screenshot showing the full chain, or a list of its factors and multiples. We hope this list inspires you to keep working on your own solutions and perhaps consider the length of the longest possible chain, and the reasons for your answer.

Let's begin with Montsaye Community College in Northamptonshire where their students took on the challenge of generating the longest possible chain.

Gabrielle and Lauren reached a total of 50 numbers:

 

Aravindan, from GHS in India, shared this chain of 51 numbers:

Makenzie, from Mountsaye Community College, managed to improve on this with a chain of 55 numbers:

 


Sophie and Tasmin, also from Mountsaye, managed to improve on that with this chain of 56 numbers:

 

 

Then, Gabrielle and Lauren managed to improve on their earlier effort with this new chain featuring 59 numbers:

 




Aravindan improved on his previous effort to produce this chain of 60 numbers:

 

Abigail from Ridgewood School managed a chain of 61 numbers:
 

Abigail's solution


Alfie, Manuel, Jack and Emilio from Newhall School in Chelmsford, Essex, worked as a team to also produce a chain of 61:

Newhall solution


A.H. from Manorfield Primary School has improved on this by finding a chain of 63 numbers:

90-9-99-33-66-11-44-22-88-8-80-40-10-100-20-60-30-15-75-25-50-5-35-7-70-14-56-28-
84-21-42-6-78-39-13-26-52-4-68-34-17-51-1-46-92-23-69-3-57-19-38-76-2-24-72-18-
36-12-48-16-24-32-96
 

This solution came in from Ralph and Max at the Institut International de Lancy in Switzerland. Their teacher said they did not use a computer or calculator. They made a chain of 65, which seems to be a very popular answer. 


Evie from Deansfield Primary School created a chain of 65 numbers:



 
James from Ridgewood School went one better with this chain of 66 numbers:
 
solution
 

Linda from Bohunt School used 68 numbers:

 

 link

A group of Year 9 students from The Perse School for Girls in Cambridge worked together and managed an even longer chain of 71 numbers:

 

 Chain of 71

 

And Claire, of Blackheath High School in London, has managed to improve on that with this chain of 73 numbers:

 


Jacky, from Princethorpe College in Rugby, has managed to go one better with a chain of 74:

 

Jesse from Moriah College in Sydney, Australia also managed a chain of 74 numbers:

 


Izaak, from Hills Rd VI Form College, used some computer programming techniques to try to make the longest chain he could. Below is his 76 long chain. If you want to read about his program and see his code, he has shared it on Github.



 

We've also received a few chains using 77 numbers!
Jakob, from Haberdashers Adams School in the UK, shared his solution:  

We received a different chain of 77 numbers from Michelle at The Coopers' Company and Coborn School in the UK:

Sarisha from Newcastle Under Lyme School, found yet another chain of 77 numbers:


Coincidentally, Ahaan, also from Newcastle Under Lyme School found the same chain but in reverse!

Well done to all of you, and to those of you who found chains of 77, congratulations, you have found the longest chain. 

Thank you to Joshua Tutin who sent us a link to The On-Line Encyclopedia of Integer Sequences, and the table below, which lists the longest possible chains using the numbers from 1 to n, where n is smaller than 200. Can you produce some of these chains?

There are many different possible longest chains - at the OEIS you can see the 12 different possible chains, of length 11 (the longest chain - see below), which use numbers from 1 to 13.

n Longest chain
1 1
2 2
3 3
4 4
5 4
6 6
7 6
8 7
9 8
10 9
11 9
12 11
13 11
14 12
15 13
16 14
17 14
18 16
19 16
20 17
21 18
22 19
23 19
24 21
25 21
26 22
27 23
28 24
29 24
30 26
31 26
32 27
33 28
34 28
35 29
36 30
37 30
38 30
39 31
40 32
41 32
42 34
43 34
44 36
45 37
46 37
47 37
48 39
49 39
50 41
51 42
52 43
53 43
54 44
55 45
56 46
57 47
58 47
59 47
60 49
61 49
62 49
63 50
64 51
65 51
66 53
67 53
68 54
69 55
70 57
71 57
72 58
73 58
74 58
75 59
76 60
77 61
78 63
79 63
80 64
81 65
82 65
83 65
84 66
85 66
86 66
87 67
88 69
89 69
90 70
91 71
92 72
93 73
94 73
95 73
96 74
97 74
98 75
99 76
100 77
101 77
102 79
103 79
104 81
105 82
106 82
107 82
108 83
109 83
110 85
111 86
112 87
113 87
114 89
115 89
116 90
117 91
118 91
119 92
120 93
121 93
122 93
123 94
124 95
125 95
126 96
127 96
128 97
129 97
130 99
131 99
132 100
133 100
134 100
135 101
136 103
137 103
138 105
139 105
140 106
141 106
142 106
143 106
144 107
145 108
146 108
147 110
148 111
149 111
150 112
151 112
152 113
153 115
154 116
155 116
156 117
157 117
158 117
159 117
160 118
161 119
162 120
163 120
164 121
165 123
166 123
167 123
168 124
169 124
170 126
171 127
172 128
173 128
174 131
175 132
176 133
177 133
178 133
179 133
180 134
181 134
182 135
183 135
184 137
185 137
186 139
187 139
188 140
189 141
190 142
191 142
192 143
193 143
194 143
195 144
196 145
197 145
198 147
199 147
200 148

 

Meanwhile, Miraya from Heckmondwike Grammar School shared these strategies for developing longer and longer chains: 

First of all not use the numbers one to ten since they are the common multiples or factors of any given number between one to hundred. Next , the other thing you have to remember is to try not to use prime numbers as the prime numbers have factors that are only 1 you can use multiples but they will not last very long. Try to find out the common multiple factors as well. Also, try to use the bigger numbers since they will have the most factors. Finally , try to use the common numbers with the most multiples under 100 for example 10 or 11 since they have multiples in every ten numbers.

Thank you, Miraya.