## Hailstones revisited

Submitted by mf344 on July 9, 2010*"Mathematics is not yet ready for such problems." This is what the mathematician John Conway reportedly said about the Collatz conjecture, a simple-looking, yet unsolved mathematical mystery. This week Plus hosted two intrepid work experience students, Sabrina Qian and Joe Dickens, who decided to have a go at this fiendish problem nevertheless. And they made some intriguing discoveries. Here is what they came up with.*

### What are hailstone sequences and what's the Collatz conjecture?

Starting with any positive integer *n*, form a sequence in the following way:

- If
*n*is even, divide by 2 to get a new number*n'*=*n*/2 - If
*n*is odd, multiply by 3 and add 1 to get a new number*n'*=3*n*+1 .

*n'*as your new starting number and repeat the process. For example,

*n*=10 gives the sequence

and *n*=25 gives the sequence

A sequence formed in this way is known as a *hailstone sequence* because the values bounce up and down just like a hailstone in a cloud. You can use the applet below to generate hailstone sequences starting with any positive integer of your choice. Enter any positive integer, the hailstone sequence will be returned.

It seems from our experiments that every hailstone sequence will eventually end up with the cycle 4, 2, 1, 4, 2, 1... and so on. Below we've plotted a few sequences starting with different numbers.

But can it be proved that *every* starting value will generate a sequence that ends up with the cycle 4, 2, 1? This is the unsolved problem known as the Collatz conjecture — no-one has so far been able to prove it.

### What can we prove?

There are, however, some things we *can* prove. First of all, we can prove that 4, 2, 1 is the only *simple* recurring pattern. A *simple* recurring pattern is one that only involves one odd number, so that the transformation 3*n*+1 is only used once. In the example 4, 2, 1 the only odd number is 1 and the only time 3*n*+1 is needed is to transform 1 into 4.

Suppose a simple recurring pattern contains the odd number *n*. Then 3*n*+1 must be a multiple of 2*n*. This is because you must be able to return to the number *n* starting from 3*n*+1 using only division by 2. In other words, we have:

*n*+1 =

*y*2

*n*,

where *y* is some positive integer.

Since *y* is a positive integer, the smallest it can be is 1. If *y*=1, then

*n*+1= 2

*n*.

*n*is a positive integer, meaning that multiplying it by 3 will give a larger answer than multiplying it by 2. If we take the next lowest value of

*y*, which is 2, we get the equation:

3*n*+1=4*n*.

Solving this for *n* gives

*n*=1.

This gives us the repeating pattern 1,4,2,1,4,2,... .

If we increase the value of *y*, so that *y* is at least 3, we get

*n*+1 =

*y*2

*n*≥ 6

*n*,

*n*,

so

*n*.

*n*is a positive integer. Thus we have proved that 4, 2, 1 is the only simple recurring pattern. If a hailstone sequence eventually settles on a simple recurring pattern, then this pattern has to be 4, 2, 1,... . However, this does not prove that there aren't recurring patterns involving more than one odd number.

Another thing that can be proved is that there are infinitely many numbers which, when "hailstoned", will produce the recurring pattern 4, 2, 1. In fact, any number *n* of the form

*n*=2

^{m},

*m*is a positive integer, will generate a hailstone sequence which eventually settles on the 4, 2, 1 cycle. Clearly such a number

*n*is even, so the first step in the hailstoning process is a division by 2 to get 2

^{m-1}. If

*m-1*=0, then 2

^{m-1}=1, so we have reached our 4,2,1 cycle. If

*m*-1 > 0, then 2

^{m-1}is even, so the next step in the hailstoning process is also a division by 2. Repeating this argument shows that all the numbers in the sequence are even, so all the steps in the process are divisions by 2, so we eventually get down to 4, then 2, then 1. This is true for any positive integer

*m*, which proves that there are infinitely many numbers which when hailstoned eventually settle on our cycle. However, this doesn't prove that

*every*number, when hailstoned, produces a sequence settling on 4, 2, 1. If this seems strange, here's a simple analogy: there are infinitely many even numbers, but that doesn't mean that

*every*number is even.

### Curiosities

When trying to start working on the problem, we looked into many areas of it, hoping to find some clues to help us when we tried to work out other things. One of the first areas we looked at was how many terms of hailstoning it took for the sequence to reach its first 4. At first the number of terms seemed fairly random, when starting with 5 it took 3 terms, when starting with 6 it took 6 terms, when starting with 7 it took 14 terms and when starting with 8 it took only one term. However, after some time it became apparent that there were some curious links between numbers.

The first thing that we noticed was that after a while some *doubles* began to appear. Doubles was the term we used when two consecutive numbers took the same number of terms to produce their first 4. The first example of this was when we hailstoned 12 and 13, which both took 7 terms to reach their first 4. This was followed by 14 and 15, which both took 15 terms to reach their first four. Although the distribution of the doubles seemed random, they consistently appeared, and before long there were also triples, quadruples and even quintuples! Below is a list of all of the doubles, triples, quadruples and quintuples that we found when hailstoning the numbers 1 to 184.

Another curiosity is that all of the doubles consist of an even number followed by an odd number (rather than the other way around), and that most of the doubles seem to come in small groups. It also seems strange that there are more quintuples than quadruples — surely it seems more likely to have 4 in a row than to have 5 in a row? And finally, why are there so many doubles, triples, quadruples and quintuples? If they appeared occasionally, they could be passed off as coincidence, but the fact that there are so many indicate that there is some underlying mathematical explanation. If you know what that might be, why not leave a comment on the *Plus* blog?

**Doubles**

12 & 13 – 7 terms

14 & 15 – 15 terms

18 & 19 – 18 terms

20 & 21 – 5 terms

22 & 23 – 13 terms

34 & 35 – 11 terms

54 & 55 – 110 terms

60 & 61 – 17 terms

62 & 63 – 105 terms

76 & 77 – 20 terms

78 & 79 – 33 terms

82 & 83 – 108 terms

84 & 85 – 7 terms

86 & 87 – 38 terms

92 & 93 – 15 terms

94 & 95 – 103 terms

114 & 115 – 31 terms

116 & 117 – 18 terms

118 & 119 – 31 terms

140 & 141 – 13 terms

142 & 143 – 101 terms

148 & 149 – 21 terms

150 & 151 – 13 terms

162 & 163 – 21 terms

180 & 181 – 16 terms

182 & 183 – 91 terms

**Triples**

28, 29 & 30 – 16 terms

36, 37 & 38 – 19 terms

44, 45, & 46 – 14 terms

49, 50 & 51 – 22 terms

65, 66 & 67 – 25 terms

68, 69 & 70 – 12 terms

108, 109 & 110 – 111 terms

124, 125 & 126 – 106 terms

145, 146 & 147 – 114 terms

156, 157 & 158 – 34 terms

164, 165 & 166 – 109 terms

172, 173 & 174 – 29 terms

177, 178 & 179 – 29 terms

**Quadruples**

Quadruples are more rare than doubles, triples, and quintuples. The first quadruple is:

**Quintuples**

130, 131, 132, 133 & 134 – 26 terms

### Further reading

The following websites contain some other methods that could be used for proving the conjecture. They are not sufficient as a complete proof, but definitely worth trying out!

- Physics Forum
- Samuel Jack
- Our sister site NRICH.