Math problem for chefs

[snousle]

A chef has received a shipment of button mushrooms that is unusually variable in size. Assume the mean size of each mushroom is 30g, with a standard deviation of 10g. In order to use these mushrooms, the chef wishes to either leave them whole, cut them in halves, or cut them in quarters, depending on their size. Devise a cutting strategy that minimizes the standard deviation of the size of the resulting pieces.

Comments

[jpeace.livejournal.com]

Blender, 40 seconds.

[dhpbear.livejournal.com]

My favorite mushrooms are of the 'magic' variety ;)

[snousle.livejournal.com]

The standard deviation is a measure of how uniform the resulting pieces are. It's not about size so much as maintaining a consistent size. It's a very commonly used thing in quality control.

[bbearseviltwin.livejournal.com]

the only one I can think of would require the step of sorting into three groups, You should end up with them roughly 1/3, 1/3, 1/3. large get quartered, medium gets halved, and the small left whole. This would seem to be rather time consuming.

[snousle.livejournal.com]

You should end up with them roughly 1/3, 1/3, 1/3

That might in fact be the answer, I don't know. It's not obvious what fraction of mushrooms goes into each pile. Getting the ideal values involves calculus. I leave it as an exercise to the reader. ;-)

[bbearseviltwin.livejournal.com]

the ratio of sizes is actually unimportant, the operative fact
is the sorting then cutting

[growler-south.livejournal.com]

But there must be optimum values for the boundaries of your three weight groups that would minimise the SD of the resultant pieces. I just don't have the math brain to calculate them.

[snousle.livejournal.com]

It's not of much practical use in everyday cooking. Although of course operations research problems like this are what make certain people billions of dollars. I just spend too much time chopping things, so my mind wanders.

[bbearseviltwin.livejournal.com]

I was just considering eye balling them by size, even I am not obsessive enough to consider actually weighing each mushroom.

[oscarlikesbugsy.livejournal.com]

The std deviation alone would be insufficient - we'd have to assume some distribution. (I suspect the larger sizes are harder to come by than a symmetrical distribution might suggest, but don't know, really).

Of course, you could patent a mushroom grader, as done here:
http://www.freepatentsonline.com/3590993.html

[growler-south.livejournal.com]

I tried to do it statistically until I remembered why I got a C+ in Statistics. Ugh. I sincerely hope you're not actually proposing to use the solution to sort and chop your mushrooms- though, knowing you, I suspect you are.

Actually, on second thoughts I have to admit it's kind of a cool problem. :-)

[oscarlikesbugsy.livejournal.com]

strategy, apart from hiring a dextrous sous chef?

Cut the little buttons in half. It increases the surface area, so they can absorb more BUTTER. Cut the big ones in quarter, using trained eye to spot those 40g+.

[chefxh.livejournal.com]

for fuck's sake, it's just some mushrooms, get back to work!

[sierrabiker.livejournal.com]

If your clients care about your deviations then that is a good thing, but if they want to make your deviations standard, then I'd say fuckum. My quality perspective.

mushroom pieces

[grizzlytx.livejournal.com]

Perhaps if you thought about what the preferred size of the resulting pieces would be? 30 gm is pretty big for a mouthful of mushroom. 15gm might be better, in which case a horseback (or heuristic, depending on how highfalutin' you want to get) strategy would be not to cut anything less than 20 grams, to halve everything between 20 and 40 grams, and to quarter the larger ones. Thus you'd have the -1sd or less mushrooms in pieces from about 10 to 20 gms (16% of the whole) the -1 to +1 sd mushrooms in 10 to 20 gm pieces (about 68% of the whole) and the +1 and beyond ones (16% also) in 10 to 13 or so gm pieces. If you wanted to halve some of the 20 gram pieces in a second round, you could do that.

This is of course not optimized, but I think it might work ok in practice. If 10-20g is too big, it would be easy to construct a similar hack for 7-10g or whatever.