Forum Discussion
LAMBDA Halloween Challenge
Finally, I have a solution!
This time I used a Lambda function to add 1 to the previous index (stored in base n) but before returning the value, I scanned the digits and copied the larger digit across.
WorksheetFormula = LET( count, COUNTA(Items), initialise, REPT("0",pick), indices, VSTACK( initialise, SCAN(initialise, SEQUENCE(combinations-1), NextCombinationλ(count,pick)) ), INDEX(Items, 1 + Explodeλ(indices)) )where
NextCombinationλ = LAMBDA(n,p, LAMBDA(initial,s, LET( place, n^SEQUENCE(1,p,p-1,-1), increment, 1 + SUM(place * Explodeλ(initial)), converted, BASE(increment,n,p), vector, Explodeλ(converted), modified, SCAN( , vector, Maxλ), CONCAT(modified) ) ) );supported by
Explodeλ = LAMBDA(term, MID(term, SEQUENCE(1,pick), 1)); Maxλ = LAMBDA(u, v, Max(u,v));I have just performed the calculation for 8 picks taken for 9 items in 300ms. Ther are 12,870 combinations and the permutations would have been over 43M!
Impressive! I'm still studying how you pulled this off. When I thought of this experiment several weeks ago, I began by studying the occurrences for each item in each column looking for patterns in the numbers.
This is the distribution of items in each column. There's some symmetry but probably not enough to go on that would top anything you've created. It's interesting that when UNIQUE is used on the above matrix, the occurrences for each number is 2 with the exception of 2 numbers appearing 4 times.
The key line is
modified, SCAN( , vector, Maxλ),which scans a row of indices to ensure each sequence increases monotonically. That allows the algorithm to make massive strides forward to the next possible combination rather than simply incrementing through the permutations and testing the outturn.