Question 1

One of Santa's best kept secrets is that there is a special link between a reindeer and its number. Indeed, a reindeer number has the property that it is equal to the product of two of the numbers (known as antlers) that can be made with its digits. For example, 1827 is a reindeer number with antlers equal to 21 and 87, because 1827 = 21 × 87. Sometimes one of the antlers gets broken (i..e. is 1 out) like 3456 is the reindeer number of a reindeer with a broken antler, because
3456 = 64 × 54 and just one digit is 1 out.

On the night before Xmas, the reindeers with these reindeer numbers reported for duty:

"It's wonderful to see you all, "said Santa, "but I can only use reindeers whose antlers are not broken. It's a long journey, and you will need all the antler power that you can get if we are to deliver all the presents."

Please help Santa sort out which of the reindeers he can use to pull his sleigh.

"Now, where is my friend Rudolf?" said Santa. "I can't go without him!" Rudolf has the smallest reindeer number and a complete set of antlers, but he hasn't reported for duty yet. What reindeer number should Santa go looking for if he is to find Rudolf?

Question 2 - Slide-together Decorations

It is something of a tradition in our house that our Christmas decorations should be home-made. Last year we found that slide-together models were really good, particularly after we had sprayed them with glue and dusted them with glitter. So we thought you might enjoy making some slide-together decorations for your classroom.

To make a slide-together, first copy the template sheets onto coloured card and cut out the triangles and squares. Next, cut along the dashed lines, making sure that you go just a tiny bit over halfway across the corner of the shape. Once you have a collection of pieces, you are ready to make a slide-together.

Join two pieces by sliding them together along the cuts that you have made, making sure that the ends join accurately. Add another piece, and continue until you have made a hole:

In these examples, we used:

Once you have made the first hole, continue adding slide-together shapes until you have made a whole 3D shape (with holes!).

The Challenge

What different slide-togethers can you make with these two shapes?

Question 3

A favourite table decoration in our house is the Star of Light. It's a five pointed start made of ten lamp holders and there are 10 lamps of various intensities (labelled 1 - 9 with 5 repeated because it is a 5-pointed star) that have to be positioned in the lamp holders.

When we first opened the box, we read the assembly instructions and can remember clever part of the decoration was that the lamps only light up when the totals of intensities along each side were the same. In the rush to clear up after Christmas last year, we lost the instructions, and now don't know where the bulbs should be positioned.

Challenge

Please give us some naturally mathematical advice as to where the bulbs should go.

If possible, could you show us what different ways the lamps can be positioned, because that's part of the charm of the decoration - it doesn't have to look the same every day.

P.S. To solve this problem, you'll find that if you click on the picture, you are linked to an applet that will be a big help in finding solutions. That's our present to you!

Question 4 - Reflection

An important part of Christmas is the looking back over the year that has past and remembering and reflecting on what happened to you and your friends.

As your last challenge for this Naturally Mathematical Challenge, we would like you, as a team, to look back over what you have done in the competition this year and to let us know:

We would really like to hear from you on these three aspects.

Diagrams for Question 2