For some reason, I'm seeing the twelve days of Christmas everywhere I look. And I don't even like that song!
Yesterday I woke up thinking about all those turtle doves, French hens, calling birds, lords-a-leaping and et cetera. While still half asleep I wondered this: if the recipient got a partridge in a pear tree the first day; then two turtle doves and another partridge and on the second; three French hens, two more turtle doves and yet another partridge on the third, etc., etc., etc...can I figure out an algebraic equation that would allow me to plug in any one of those gifts and quickly figure out how many were given in total, over the twelve days of rampant gift-giving?
I do love puzzles, and I was happy to have stumbled across one to wrap my brain around before my feet even hit the ground.
Try it, Puzzleheads! Or try it with your algebra students. It's definitely doable, and not too hard. The answer is in the comment section following this post.
If you want to work it out on your own, stop here. If you want some help, read on. If you think the whole thing is totally nuts, go bake some cookies and bring me some. I'll put the coffee on.
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How to solve:
Let T be the total number of any certain thing.....laying geese, golden rings, whatever. T is the total number of that particular gift in hand after the twelve days expire.
Let D stand for the day on which that gift was first introduced, or, to phrase it a little differently, D can stand for how many of that particular gift were given at a time. They are the same number. So, for maids-a-milking, D = 8, because eight maids-a-milking first show up on the 8th day of Christmas, and then eight are given each day through the 12th..
Now comes the thinking part. Obviously, how many swans-a-swimming you get will be the number of swans you receive per day, times the number of days you receive them. Think about how many days you get turtle doves, and how many days you get French hens, etc., and see if you can find a pattern that relates the quantity of how many you get at a time to the number of times, total, that you receive that quantity. If you can make that pattern into an equation, you're home free. Try out your formula on a couple of different gifts and see if it works. If it does, you've solved the puzzle!
Congratulations. And don't forget to send me the cookies!
2 comments:
Remember that the total (T) = the number of drummers (or whatever) you receive each day, times the number of days you receive them. If you pencil it out, you see that you get 1 partridge on 12 different days, 2 turtle doves 11 times, 3 French hens 10 times, 4 calling birds 9 times, etc.
The number you receive on any given day, plus the number of times you receive it, always adds up to 13! There's your pattern.
So if you subtract D (the number you receive on any given day) from 13, you're left with a number telling how many days you got that gift. Multiply that times D and you'll have your total.
D x (13 - D) = T
And, by the way, you get 42 geese and 42 swans - more than any of the other gifts. Just FYI.
OMG, the things you think of. If you really want to have some fun, look into the golden ratio.
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