Math CentralQuandaries & Queries


Question from Kathy:

I have 680 champagne glasses and my pyramid can only be 15 levels high. How many glasses need to be on each level? Is there a formula I can use?


The levels will be triangular numbers 1,3,6,10,... and the totals will be "tetrahedral numbers' 1,4,10,20,... At a height of 15 you will get exactly 680 glasses in. Your biggest layer will be a triangle of (15*16)/2 = 120 glasses, in tight-packed rows of 15,14,...,2,1. After that, each layer goes exactly over the holes in the one below.

This may take an inordinate time to fill; at 6 glasses to the bottle it will take over 110 bottles, which could take at least half an hour to pour, by which time the wine will be going flat (especially in a coupe, which is not the ideal champagne glass.) Magnums or jeroboams will look better but won't improve the time much.

If you try to be a purist & pour only from the top, the glasses in the corners will never fill until the ones in the middle have been overflowing for a long time. Even two layers down the three corner glasses will only be 1/3 full when the others are full. In each layer, the corner glass gets 1/3 as much as the one above it gets. Thus the corner glass in the nth layer gets full when the glass above it has been filled and another 3^n glassfuls have been poured! To fill every glass in a 15-high fountain with a center pour would take 1+3+9+...+314 = (315 - 1)/2 glassfuls, or over a MILLION bottles! I don't imagine the budget for the event - or the guests' patience - would stretch to that.

Make sure the pourer[s] don't keep pouring into the top glass only. Once the bottom center glass starts to overflow the pourer should start pouring down the three ridges; and when the bottom glasses start to overflow anyhow they should start serving, topping off any light glasses. If the number of guests is significantly less than the number of glasses they should have some idea how many glasses in the corners to leave dry.

My advice would be to have a row of smaller fountains (maybe five of them, eight high, with 120 glasses in each) being filled simultaneously. But that's up to you.

Also, as many of the glasses will be completely full, you should find out how many completely full glasses there are to the bottle for this particular make of glass. This may affect how many bottles you need to buy; if many of the portions are closer 1/4 bottle you will need more wine.

Good luck!


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