Which is faster — sorting or multiplying a small array of elements?

Of course it depends a lot on the CPU of your computer, but a typical Intel CPU (e.g. Core 2 Duo) can multiply two 32 Bit numbers within 3 CPU clock cycles. For a sort algorithm to beat that, the algorithm needs to be faster than 3 * 4 = 12 CPU cycles, which is a very tight constraint. None of the standard sorting algorithms can do it in less than 12 cycles for sure. Alone the comparison of two numbers will take one CPU cycle, the conditional branch on the result will also take one CPU cycle and whatever you do then will at least take one CPU cycle (swapping two cards will actually take at least 4 CPU cycles). So multiplying wins.

Of course this is not taking the latency into account to fetch the card value from either 1st or 2nd level cache or maybe even memory; however, this latency applies to either case, multiplying and sorting.

Without testing, I'm sympathetic to his argument. You can do it in 4 multiplications, as compared to sorting, which is n log n. Specifically, the optimal sorting network requires 9 comparisons. The evaluator then has to at least look at every element of the sorted array, which is another 5 operations.

Sorting is not intrinsically harder than multiplying numbers. On paper, they're about the same, and you also need a sophisticated multiplication algorithm to make large multiplication competitive with large sort. Moreover, when the proposed multiplication algorithm is feasible, you can also use bucket sort, which is asymptotically faster.

However, a poker hand is not an asymptotic problem. It's just 5 cards and he only cares about one of the 13 number values of the card. Even if multiplication is complicated in principle, in practice it is implemented in microcode and it's incredibly fast. What he's doing works.

Now, if you're interested in the theoretical question, there is also a solution using addition rather than multiplication. There can only be 4 cards of any one value, so you could just as well assign the values 1,5,25,...,5^12 and add them. It still fits in 32-bit arithmetic. There are also other addition-based solutions with other mathematical properties. But it really doesn't matter, because microcoded arithmetic is so much faster than anything else that the computer is doing.