A truly uniform distribution, no. (Well, your question is not completely well posed, but I will argue this for most ways of making it so.) There are $n!$ (kn)!$factorial ways to shuffle a deck of$(kn)!$kn$ cards. So you want each permutation to occur with probability $1/n!$. 1/(kn)!$. In particular, for every prime$p \leq kn$, you want$p$to occur in the denominator of the probability that each permutation occurs. Let's look at your operations: Reordering$i$decks can only introduce primes$\leq i$. Perfectly shuffling an$n$card deck can only introduce primes$\leq n$. Cutting depends on what mathematical model you use for cutting; if all cut points are equally likely, you only get primes dividing$n(n-1)\ldots (n-i+1)$. I imagine other models of cutting will cause similar problems. The more commonly studied question is how to get a probability distribution that is extremely close to random. There are lots of good results on this; see Trailing the Dovetail Shuffle to its Lair. 1 A truly uniform distribution, no. (Well, your question is not completely well posed, but I will argue this for most ways of making it so.) There are$n!$factorial ways to shuffle a deck of$(kn)!$cards. So you want each permutation to occur with probability$1/n!$. In particular, for every prime$p \leq kn$, you want$p$to occur in the denominator of the probability that each permutation occurs. Let's look at your operations: Reordering$i$decks can only introduce primes$\leq i$. Perfectly shuffling an$n$card deck can only introduce primes$\leq n$. Cutting depends on what mathematical model you use for cutting; if all cut points are equally likely, you only get primes dividing$n(n-1)\ldots (n-i+1)\$. I imagine other models of cutting will cause similar problems.