mathematics

Wichmann–Hill

Wichmann–Hill is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Wichmann–Hill rather than just read about it. In short: Wichmann–Hill is a pseudorandom number generator proposed in 1982 by Brian Wichmann and David Hill. It consists of three linear congruential generators with different prime moduli, each of which is used to produce a uniformly distributed number between 0 and 1.

Key takeaways

Wichmann–Hill belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Wichmann–Hill to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Wichmann–Hill from memory before moving on to harder problems.

Reference excerpt

Wichmann–Hill is a pseudorandom number generator proposed in 1982 by Brian Wichmann and David Hill. It consists of three linear congruential generators with different prime moduli, each of which is used to produce a uniformly distributed number between 0 and 1. These are summed, modulo 1, to produce the result. Summing three generators produces a pseudorandom sequence with cycle exceeding 6.95×1012. Specifically, the moduli are 30269, 30307 and 30323, producing periods of 30268, 30306 and 30322. The overall period is the least common multiple of these: 30268×30306×30322/4 = 6953607871644. This has been confirmed by a brute-force search.

Implementation The following pseudocode is for implementation on machines capable of integer arithmetic up to 5,212,632:

[r, s1, s2, s3] = function(s1, s2, s3) is // s1, s2, s3 should be random from 1 to 30,000. Use clock if available. s1 := mod(171 × s1, 30269) s2 := mod(172 × s2, 30307) s3 := mod(170 × s3, 30323)

r := mod(s1/30269.0 + s2/30307.0 + s3/30323.0, 1)

For machines limited to 16-bit signed integers, the following equivalent code only uses numbers up to 30,323:

[r, s1, s2, s3] = function(s1, s2, s3) is // s1, s2, s3 should be random from 1 to 30,000. Use clock if available. s1 := 171 × mod(s1, 177) − 2 × floor(s1 / 177) s2 := 172 × mod(s2, 176) − 35 × floor(s2 / 176) s3 := 170 × mod(s3, 178) − 63 × floor(s3 / 178)

r := mod(s1/30269 + s2/30307 + s3/30323, 1)

The seed values s1, s2 and s3 must be initialized to non-zero values.

References

Worked examples

Example 1 — a first encounter with Wichmann–Hill

Start with the simplest possible case. Write down what Wichmann–Hill claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Wichmann–Hill before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Wichmann–Hill ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Wichmann–Hill

In research: Wichmann–Hill appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Wichmann–Hill in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Wichmann–Hill is common in secondary-school and first-year university syllabi. It links to neighbouring topics Pseudorandom number generators, so understanding it makes those chapters shorter.
In everyday life: Look for Wichmann–Hill outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Wichmann–Hill in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Wichmann–Hill means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Wichmann–Hill out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Wichmann–Hill in simple terms?

Why does Wichmann–Hill matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Wichmann–Hill?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Wichmann–Hill.