## Time Trial on Trust

I have a selection of 4 fairly easy puzzles. You can do these in your head. Time yourself, and, if you dare – publish your time along with the answers. Read the questions carefully, and your time starts……. NOW!

1. A large water tank has two inlet pipes (a large one and a small one) and one outlet pipe. It takes 2 hours to fill the tank with the large inlet pipe. On the other hand, it takes 5 hours to fill the tank with the small inlet pipe. The outlet pipe allows the full tank to be emptied in 7 hours. What fraction of the tank (initially empty) will be filled in 1.35 hours if all three pipes are in operation? Give your answer to two decimal places (e.g., 0.25, 0.5, or 0.75).

2. The son of a rich bullion merchant left home on the death of his father. All he had with him was a gold chain that consisted of 98 links. He rented a place in the city center with a shop at the lower level and an apartment at the upper level. He was required to pay every week one link of the gold chain as rent for the place. The landlady told him that she wanted one link of the gold chain at the end of one week, two gold links by the end of two weeks, three gold links by the end of three weeks and so on. The son realized that he had to cut the links of the gold chain to pay the weekly rent. If the son wished to rent the place for 98 weeks, what would be the minimum number of links he would need to cut?

3. A cylinder 48 cm high has a circumference of 16 cm. A string makes exactly 4 complete turns round the cylinder while its two ends touch the cylinder’s top and bottom. How long is the string in inches?

4. My Dad has a miniature Pyramid of Egypt. It is 3 inches in height. Dad was invited to display it at an exhibition. Dad felt it was too small and decided to build a scaled-up model of the Pyramid out of material whose density is (1 / 5) times the density of the material used for the miniature. He did a “back-of-the-envelope” calculation to check whether the model would be big enough. If the mass (or weight) of the miniature and the scaled-up model are to be the same, how many inches in height will be the scaled-up Pyramid?

July 25th, 2014 at 7:31 pm

1. If 1.35 hours means 1 7/20 hours then 0.75. If it means 1 hour 35 minutes then 0.86.

2. 6 cuts.

3. 112 cm.

4. 6.71 inches.

How long? About 15 hours (including sleep, meals, TV, etc).

July 25th, 2014 at 8:31 pm

No way can I do 1 or 4 in my head.

1. 0.75(21) full (assuming constant emptying rate and 1 h 21 m)

2. 4 cuts. Why 98 rather than 159? What have I missed?

3. 80 cm

4. 5.1 inches

25 mins

July 26th, 2014 at 1:45 am

To elaborate:

1. In 1 hour tank will 1/2 fill from the large pipe, 1/5 fill from small pipe and 1/7 empty from outlet pipe. So proportion of tank filled after 1.35 hours is (1/2 + 1/5 -1/7) * 1.35 = 0.75217285…

2. Cut successive links of length 1, 2, 4, 8, 16 and 32. By swapping links of appropriate length with the landlady these 6 cuts can pay the first 63 days rent. The remaining 35 links on the chain can then be combined with the others as required to get up to 98.

3. If the string is wound horizontally around the cylinder then four times round needs 64 cm plus a further 48 cm for the ends to reach top and bottom. If the string is wound spirally from (say) bottom to top then if we are looking for the MINIMUM string length (which is not stated in the problem) then the length required is the hypotenuse of a right angled triangle with short sides of 48 and 64 cm, which is 80 cm.

4. I now agree with Chris! The new model is 5 times larger than the original, so each dimension, including the height, increases by a factor of the cube root of 5, i.e. 3 * 1.71 = 5.13. I was erroneously using the square root.

July 26th, 2014 at 9:54 am

Q2. This isn’t like you Wiz. You’ll award yourself a slap-head when you see why 4 cuts is enough.

As for the choice of 98 links. This is a Karl wind-up number.

Q3. Correction: 30.496 (approx.)

July 26th, 2014 at 8:46 pm

Uh oh! Q3 in cm, answer in inches (actually 80 cm = 31.496 inches). Sneaky!

July 27th, 2014 at 4:41 am

Ooops. I don’t know how I ended up typing 30.496 rather than 31.496.

Now prepare to slap head.

Cut 4 links to get e.g. 1,1,1,1,5,10,20,40,19. The 1’s are the cut links.

July 27th, 2014 at 10:43 pm

If the chain was a loop, then 5 cuts are needed.

1, 1, 1, 1, 1, 6, 12, 24, 48, 3 does it.

Still looks like Karl’s 98 is a wind-up number.

July 28th, 2014 at 12:27 am

I take it that the 98-link chain looks something like this:

5 1 10 1 20 1 40 1 19

where the 1’s denote the positions of the links to be cut.

Self-slap duly administered. I’m pretty sure I’ve seen this somewhere before which makes my lapse even less excusable.

I’ve no idea why 98 is the chosen number of links. As you say, it could be a lot more (replace the 19 by 80 to get 159).

July 28th, 2014 at 6:59 am

Qu.1 – 0.75.

Qu.2 – If the mass is to be the same, then density is inversely proportional to volume. Also, the volumes are directly proportional to the cubes of the heights for objects that are geometrically similar. Therefore, the heights are seen to be inversely proportional to the cube roots of the densities. Thus, Height of model = 3 × 51/3 = 5.13 inches.

Qu.3 – The whole string measures 80 cm, But the question asks for the answer in inches. Bad of me I know, but I am hoping to catch some of you out. 31.496063 Inches

Qu.4 – Note that when a link in the center of the chain is cut, three pieces are obtained: a one-link piece and two other pieces. For example, when the third link in a chain consisting of 6 links is cut what is obtained is a one-link piece, a two-link piece and a three-link piece.

The minimum number of cuts needed to be made is 4 for a chain with 98 links.

If the links are numbered serially from 1 to 98, then the cuts would be made on the following links:

6, 17, 38, and 79. Better expressed by Wiz in Post 8 as – 5 1 10 1 20 1 40 1 19

This would result in 4 one-link pieces, one 5-link piece, one 10-link piece, one 20-link piece, one 40-link piece, and one 19-link piece.

To gain a better understanding, consider the scenario in the first few weeks as illustrated in the table below.

Week: 1 2 3 4 5 6 7 8 9 10 11 12

Gold links given:

1

1+1

1+1+1

1+1+1+1

5

5+1

5+1+1

5+1+1+1

5+1+1+1+1

10

10+1

10+1+1

The table above indicates that:

-at the end of the fifth week, the 5-link piece is given and the 4 one-link pieces are taken back;

-at the end of the tenth week, the 10-link piece is given and the 4 one-link pieces as well as the 5-link piece are taken back.

Why 98 – because it made you think there was something more to the question?