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Cubic Conundrum

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Cubes

Stage: 1 and 2 Challenge Level: Challenge Level:1

Before you read much further I think it will help if you get yourself three cubes that are the same size. Maybe if you are at home they could be the kind that very young toddlers have to play with, made from wood or plastic. If you are at school it should be no problem, but avoid those cubes that have bits for connecting together. Just three cubes of the same size.


pic 1

Place just one of these cubes down in front of you on the table. I'm going to ask a question that gets repeated each time we try a new idea. Here's the question:

"If you are allowed to walk around the table, bend down low, move your head around in any way that you like, get as near to the cubes as you like, etc. then how many square faces could you see altogether, without lifting up or touching the cubes in any way?''

Well I guess that with one cube it is quite easy, yes it's 5.

Now use two cubes and put them in any way that you like. Well, here are three arrangements I did earlier.

pic 2

The first one lets me see 9, the second one 8 and with the two on their own it's 10! So the answer to the question depends on how the cubes are arranged. I suppose that's pretty obvious. Well, what about three cubes? Here are some arrangements - just some.

pic 3

The first has 11; the second has 13; the 3 on their own have 15; the last one has 14 and two-thirds. Now that's a strange one - you can see why we need "non-connecting'' cubes. "How did I get 14 and two-thirds?'' you may be asking. It's the sort of question I ask the children I'm working with. I'll tell you. The red shows 5 whole faces, the green and the blue each show 4 whole faces. There is about 1/3 of the underneath of the red I could see if I creep down low and look up when I'm very close. There is 2/3 on top of both the blue and the green that are visible. That makes a total of 14 and 2/3.

So you take your 3 cubes and see what your answer to the question is each time you make a new arrangement. It's probably good to record your answers. This is very easy with some art packages on the computer. You might like to get an arrangement that produces a particular number of faces visible. See what your largest/smallest answer is. When you think you have got sufficient answers, probably something in the region of 10, then use a fourth cube and see how you get on now. Good Luck.