The “bean-feast” is a European phrase used primarily for an annual dinner given by an employer for their employees. Originally, it was a custom in France, and then later in Germany and England, of a feast on Twelfth Night, which is the eve after Epiphany. A cake was created with one bean in it. The person who received the slice with the bean was the king or queen of the festivities of Twelfth Night. There are some cultural variants that state the king or queen was also to perform ceremonies to ensure good weather for the twelve months during the year.
Another variation of this type of cake is the King Cake, which is currently a tradition of the Mardi Gras season in New Orleans. It is usually round or oval, and has a baby hidden inside, although some stories state that it was once a bean. The person that receives the baby is the king or queen for the night’s festivities. They can also pick a royal partner for the occasion.
Here is a Victorian puzzle represented in the book Amusements in Mathematics by Henry Dudeney, entitled “The Beanfeast”, which can be found at Project Gutenburg. Some of the puzzles within the book were reprints from popular magazines of the time, such as The Strand Magazine, Cassell's Magazine, The Queen, Tit-Bits, and The Weekly Dispatch.
“A number of men went out together on a bean-feast. There were four parties invited—namely, 25 cobblers, 20 tailors, 18 hatters, and 12 glovers. They spent altogether £6, 13s. It was found that five cobblers spent as much as four tailors; that twelve tailors spent as much as nine hatters; and that six hatters spent as much as eight glovers. The puzzle is to find out how much each of the four parties spent.” – Henry Dudeney, 1917