1. HOW MANY BEARS?    

Here is an old puzzle you might like to solve and then try out on your friends. Mr Cleverwun tosses a pair of dice and asks, “How many polar bears are there around the ice hole?”

At right are some sample throws with answers provided by Mr C:

Now what about the throw shown at bottom?

Hint 1

Hint 2

Solution

Extensions

Hint 1

Have I made use of  all the information?

Hint 2

Make a systematic list of the throws and answers.

Can you spot a pattern?

Solution

We expect there to be some rule which will give us the answer each time from the numbers appearing on the dice. We observe that the given answers are None, Two, Four and Six. So let us collect together those throws giving the same answer. We obtain:

None:  4, 2; 1, 1; 1, 4; 6, 6; 4, 4;
Two:    3, 6; 2, 3;
Four:   3, 3; 1, 5; 6, 5; 5, 2;
Six:     3, 5.

We have now assembled the information, and it is now that the Aha! must come. Look at this list. Which numbers are scoring numbers?

From the first row it appears that 1, 2, 4, 6 make no contribution. From the second row, each 3 contributes two to the score. This checks out for the first pair in the third row. Now each 5 contributes four, and the fourth row confirms this. Remembering the title, only the die-faces with an ice-hole in the middle contribute; that is, the 3 and the 5, and the score is the number of dots surrounding the central dot. Thus the required answer for a throw of two fives is eight.

Extensions

1. You might like to use your imagination to devise some different questions for the throw of two dice, corresponding of course to different scoring rules. For example: ‘How many leaning towers ar there in Pisa?’ (score one for each 2 and one for each 3), or ‘How many types of animal in the ark?’ (score half the dots on each even die face).

2. We are making an assumption that the answer’ is in fact related in some way to the score of the dice. I remember an old party game where the answer to ‘Are these scissors crossed or uncrossed?’ had more to do with the position of the questioner’s legs than the scissors themselves!