Global Lotto Index
Simulator
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The Index Mode

Six numbers, one index.

Every valid lottery ticket can be assigned a unique number from 1 to C(n,k). Sort the combinations lexicographically; the position is the index. That's the whole trick.

Worked example · 6/49

#1
#2
#1,000,000
#7,654,321
#13,983,816

Why it matters

  • Compact. Save a ticket as one number. Share it as a link parameter. Backtest a million plays with bigint arithmetic.
  • Reversible. rank(unrank(i)) = i, always. We test it with property tests.
  • Uniform. Sampling a random index is sampling a uniformly-random ticket — no bias from naive number-by-number picking.
  • Analyzable. Index distributions reveal structure: prime indexes, parity, clusters near the high or low end of the space.

The algorithm

We use the combinadic representation. To rank a sorted combination c1 < c2 < … < ck, count how many combinations come lexicographically before it — that's a sum of binomial coefficients. Unranking inverts the sum greedily. Complexity is O(k · (n−k)).

rank(c, n, k):
  idx = 0; prev = 0
  for i in 0..k-1:
    for v in prev+1 .. c[i]-1:
      idx += C(n - v, k - i - 1)
    prev = c[i]
  return idx + 1   // 1-based
Try it in the simulator