How does the mid-square method get the next number?

Squares the seed and takes the middle digits

Multiplies the seed by a constant and takes the remainder

Adds the last two numbers and takes mod m

Reads a table of random numbers

Why is the mid-square method considered weak?

It may degenerate quickly into a short repeating cycle

It always gives numbers above 0.5

With a = 17, c = 43, m = 100, X0 = 27, what is X1 and what does it show?

X1 = 2; the sequence has a short period of only 4

X1 = 77; it has full period 100

2073 Bhadra · 8 marks

Random number generation methods

Short answer

Random numbers are generated by methods such as mid-square, mid-product, constant multiplier, linear congruential X(n+1)=(aX(n)+c) mod m, multiplicative, additive, and hardware sources. Each produces pseudo-random sequences that must be uniform, independent, fast, and have a long period.

What are the different random number generation methods? Explain with examples.

Random numbers for simulation are produced by a deterministic formula (pseudo-random numbers) that gives a sequence that looks random. Main methods:

1. Mid-square method (von Neumann)

Square the seed and take the middle digits as the next number.

Seed 1234 (4 digits): 1234^2 = 01522756, middle four digits give 5227, so R1 = 0.5227. 5227^2 = 27321529 gives 3215 (R2 = 0.3215), then 3362, 3030, 1809 ...

Weakness: the sequence may degenerate quickly. Seed 2100: 2100^2 = 04410000 gives 4100, then 8100, 6100, 2100 and repeats after 4 numbers.

2. Mid-product method

Multiply two successive numbers and take the middle digits. With x0 = 5731 and x1 = 7182: 5731 x 7182 = 41160042, middle 1600; then 7182 x 1600 = 11491200, middle 4912; then 1600 x 4912 = 07859200, middle 8592.

3. Constant multiplier method

Same as mid-product, but every number is multiplied by a fixed constant a. With a = 7182 and x0 = 5731: 7182 x 5731 = 41160042 gives 1600; 7182 x 1600 = 11491200 gives 4912; 7182 x 4912 = 35277984 gives 2779.

4. Linear congruential method (LCG)

X(n+1) = (a X(n) + c) mod m, R(n) = X(n) / m.

Example: a = 17, c = 43, m = 100, X0 = 27. X1 = (17 x 27 + 43) mod 100 = 502 mod 100 = 2; X2 = (34 + 43) mod 100 = 77; X3 = 52; X4 = 27, then it repeats (period 4 only, because the parameters were poorly chosen).

Better example: a = 5, c = 3, m = 16, X0 = 7 gives 6, 1, 8, 11, 10, 5, 12, 15, 14, 9, 0, 3, 2, 13, 4, 7: full period 16. R = 0.375, 0.0625, 0.5 ...

The multiplicative form (c = 0) is X(n+1) = a X(n) mod m. Example: a = 13, m = 64, X0 = 1 gives 13, 41, 21, 17, 29, 57, 37, 33.

5. Additive congruential method

X(n) = (X(n-1) + X(n-k)) mod m. With X(n) = (X(n-1) + X(n-2)) mod 100 and seeds 65, 89 the sequence is 54, 43, 97, 40, 37, 77.

6. Other sources

Library functions such as rand() (usually an LCG).
Tables of random numbers (RAND table), now rarely used.
Hardware (physical) generators using noise or radioactive decay, which are truly random but cannot be repeated.

A good generator has a long period, uniform output and no correlation, and is checked with the chi-square, K-S, runs, gap and poker tests.

✓ Answer checked by Claude Opus 5.5 (high effort)

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