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Friday 30 April 2021

Sums of Cubes

 This puzzle comes from a tweet by Ed Southall (@EdSouthall), who identified the following mathematical relationship:

3³ + 5³ + 2³ = 160 1³ + 6³ + 0³ = 217 2³ + 1³ + 7³ = 352 3³ + 5³ + 2³ = 160

The question (not posed, but I'm answering it anyway) is: Is this a unique relationship, or are there more like it?

And the answer is that there are a lot more like it.  I have carried out a brief search and found a total of eight specific groups.  Here they are, in the order that I discovered them (starting with 100, the first three-digit number).

Terminate at 1 

1³ + 0³ + 0³ = 1

1³ = 1

It's not exactly earth-shattering, but here's the first case:  terminates at 1.  Example numbers which do this are: 100, 112, and 121.

Terminate at 153

1³ + 0³ + 2³ = 9

9³ = 729

7³ + 2³ + 9³ = 1080

1³ + 8³ = 513

5³ + 1³ + 3³ = 153

1³ + 5³ + 3³ = 153

Other numbers which do this are: 108, 111, and 135.  The example of 102 goes via 1080, while 105 follows a longer route:

1³ + 0³ + 5³ = 126

1³ + 2³ + 6³ = 225

2³ + 2³ + 5³ = 141

1³ + 4³ + 1³ = 66 6³ + 6³ = 432 4³ + 3³ + 2³ = 99 9³ + 9³ = 1458 1³ +4³ + 5³ + 8³ = 702 7³ + 2³ = 351

3³ + 5³ + 1³ = 153

1³ + 5³ + 3³ = 153

Loop with a low of 55

1³ + 0³ + 3³ = 28

2³ + 8³ = 520 5³ + 2³ = 133

1³ + 3³ + 3³ = 55
5³ + 5³ = 250
2³ + 5³ = 133

Terminate at 371

1³ + 0³ + 4³ = 65
6³ + 5³ = 341
3³ + 4³ + 1³ = 92
9³ + 2³ = 737
7³ + 3³ + 7³ = 713
7³ + 1³ + 3³ = 371
3³ + 7³ + 1³ = 371

Terminate at 370

1³ + 0³ + 9³ = 730
3³ + 7³ + 0³ = 370
3³ + 7³ + 0³ = 370

Loop around 160  (the example given in the question)
1³ + 1³ + 5³ = 127
1³ + 2³ + 7³ = 352
3³ + 5³ + 2³ = 160
1³ + 6³ =217
2³ + 1³ + 7³ = 352

Loop around 136 and 244  
1³ + 3³ + 6³ = 244
2³ + 4³ + 4³ = 136

407: Identity
4³ + 0³ + 7³ = 407

Summary


The eight conclusions of the 'cubic sums' puzzle for three-digit numbers (which I found) are: 
Terminate at 1
Terminate at 153
Loop around 55  (55, 250, 153, 55) 
Terminate at 371
Terminate at 370
Loop around 160 (160, 217, 352, 160)
Loop around 136 and 244  
407 - identity

There are probably other 'identities' but there's no pattern to them and they would only be found by trial and error (or an algorithm).

Summary of results for 101 to 151

Start Result
101 371
102 153
103 Loop 55
104 371
105 153
106 Loop 106
107 371
108 153
109 370
110 371
111 153
112 1
113 371
114 153
115 Loop 160
116 371
117 153
118 370
119 371
120 153
121 1
122 371
123 153
124 370
125 371
126 153
127 Loop 160
128 371
129 153
130 Loop 55
131 371
132 153
133 Loop 55
134 371
135 153
136 Loop 136
137 371
138 153
139 370
140 371
141 153
142 370
143 371
144 153
145 370
146 371
147 153
148 370
149 371
150 153
151 Loop 160


It's interesting (and probably coincidental) to note that 137 to 150 show a repeating pattern of their own - the results go through a sequence 371, 1543, 370).  This is probably only a coincidence, but it's interesting how these results crop up!


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