Neurobics

A Selection of Problems

Russell Dear

The First Day Of A Century

On what days of the week can the first day of a century fall?

The Lost Coin

After having her dollar changed into six coins, a girl lost one. What is the probability that it was a ten cent piece?

Pets

Chloe, Simon, Amy, and Gemma are the cause of much confusion at home. Each of them owns an axolotl named after one of the other three and a parakeet named after another of them. No two axolotls, and no two parakeets, have the same name.

For example, Gemma's parakeet and Amy's axolotl are both namesakes of the owner of the axolotl Amy. The namesake of Simon's axolotl is the owner of the axolotl whose namesake owns the parakeet Chloe.

Who owns the parakeet Gemma?

Loading problem

Two bins, A and B, each of capacity 20 units are to be loaded with items. Each item is of integral size from one to nine units. The items arrive singly at random and each is loaded before the next is sighted.

For example, suppose items of size 5, 9, 7, 8, 4, 6, 5, arrive then the 5 and 9 could be loaded in A but the next two items, the 7 and 8, would have to be placed in B since there is no room in A. At this stage A contains 14 and B 15. The next item to be loaded is 4. It can go in either bin. If it is placed in A then the next item 6 is too big for either bin and the packing is complete with wastage two in A and five in B, a total of seven. Alternatively, if the 4 had been placed in B then there would have been room in A for the next item, 6, and a total wastage of only one would have occurred.

Which packing strategy is the most efficient, that is, which on average gives the least wastage?

Odd Triangles

Believe it or not, these two triangles have the same perimeter and area.

Now look at these two triangles and decide which has the largest area.

The Pool Balls

Colleen and Reg were playing pool one lunchtime. They placed the balls, which are numbered from 1 to 15, in a triangle at random (three are shown in the diagram).

Just as he was about to begin play Colin exclaimed, "Well, look at that. The number on every ball is the difference of the numbers on the two balls immediately above it." (except for the balls in the top row, of course, which don't have any balls above them).

How were the fifteen balls arranged?

Answers

1) Only on Monday, Tuesday, Thursday, or Saturday. One proof rests on letting Sunday = 1, Monday = 2, and so on, and looking at alternatives in arithmetic modulo 7.

2) There are only three ways a dollar can be changed into six coins: 50, 10, 10, 10, 10, 10 or 50, 20, 10, 10, 5, 5 or 20, 20, 20, 20, 10, 10 cents. Hence the probability of the lost coin being 10c is 1/35/6 + 1/32/6 + 1/32/6 = 1/2

3)A systematic approach or trial-and-error should give the correct solution that the parakeet Gemma is owned by Amy.

4) The one I've found to be most efficient, with a mean wastage of approximately 3.9 units, places all items in bin A until one won't fit, then try that one in B. Repeat the procedure until an item will not fit in B -- then stop. You may be able to do better -- let me know. By the way, did you know that there are three types of mathematician? Those that count, and those that don't!

5) Surprisingly, even if the numbers on the three balls had not been given there is only one solution, and its reflection, to this problem. From top to bottom, left to right, the balls are numbered 13, 3, 15, 14, 6, 10, 12, 1, 8, 2, 11, 7, 9, 4, 5.

6) Believe it or not the triangle on the left has the larger area of 240cm2 compared with that on the right with area 234cm².

Russell Dear is a Mathematician living in Invercargill