The Hailstone Program is a tool for investigating the behavior of numbers when they are exposed to the
Hailstone process defined in the
Introduction. It forms part of the program
Crossword Express which is a free-ware program you can download from
the Download page of the Crossword Express web site. The Download page includes extensive instructions
to assist you with downloading, installing and operating the program. Please study these instructions very thoroughly,
and follow them to the letter.
When you have completed the instructions for Starting Crossword Express, you will see the following which is a
slightly reduced image of the Crossword Express Control Center:-
The selection box provides access to all of the functions of Crossword Express, including Hailstone. Select Hailstone
and click and GO to display this Hailstone screen.
As you can see, the screen consists mainly of two text areas labeled
Number and
Signature, and that they
are already populated with the number 27 and its Signature. There is plenty of room for such a small number and
Signature, but the program is capable of processing vastly larger examples. To go some way toward accommodating such
examples, the screen is resizable and relocatable in the usual way.
The two buttons in the center of the screen
provide a simple means of deriving a Signature from a number or a number from a Signature.
- Enter 27 into the Number area, and click the Number to Signature button. The Signature will appear in the
Signature area and you will also see a dialog which contains some additional useful information. (Remember that a
Signature Chunk starts with an O, and ends immediately before the next O):-
- Test the reverse operation of converting a Signature back to the original number by clicking the Signature to
Number button. The result which appears in the Number field may surprise you. It is:-
2361183241434822606848n + 27
If n is 0, then the answer is 27 as expected. Otherwise it is 27 plus a multiple of that 22 digit number. So,
there is a never-ending series of numbers, all with Signatures which are much longer than that of 27, but all of
which start with the same 112 letters.
Make a point of studying the contents of the dialog. What you will see is a line by line description of the steps
taken by the algorithm as it computes the number. Some of the numbers in this description become inconveniently
large, and when this happens you will see entries like this [18 digits]n+122 which should be self
explanatory. Note in particular the way in which the final answer is assembled by the timely addition of powers of
two.
- Amuse yourself by typing ever larger numbers into the Number field, and observing the resulting Signatures.
- In case you doubt the accuracy of the program, you could generate the Signature of any number that comes to mind
using a calculator or even pencil and paper, and type that string into the Signature field. Clicking Signature
to Number, should return your original number into the Number field.
- Create your own random Signatures, and see what sort of number is returned using the Signature to Number
button. Remember the Signature must begin with O, it must contain only Es and Os, but must not contain
consecutive Os.
- The Hailstone Program also includes a batch processing mode in which hundreds (or even tens of thousands) of
numbers are processed en-masse and the results combined to provide a much more accurate impression of what is
going on. Details of this mode, and instructions on how to use it are presented
at The Hailstone Rule-of-8 Demonstration.