A sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.

The first 7 terms would be: 1, 3, 6, 10, 15, 21, 28

Let us list the factors of the first seven triangle numbers:

1: 1

3: 1,3

6: 1,2,3,6

10: 1,2,5,10

15: 1,3,5,15

21: 1,3,7,21

28: 1,2,4,7,14,28

We can see that 28 is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

Tags:

Maths
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?

Tags:

Maths
Every day, a large number of people commute from city A to D by going through B and C. There are 4 roads

and the travel times are

A-B 40 minutes

C-D 40 minutes

The travel times on the two other roads depend on the proportion of traffic that uses them. Let p be the proportion of the commuters that use a segment.

A-C 30*p minutes

B-D 30*p minutes

For example, if 90% of the traffic goes A-C then it takes them 27 minutes.

An equilibria is reached with p = 50%, and each commute requires a total of 40 + 15 = 55 minutes on both paths A-B-D and A-C-D.

Here is the question: Construction a new road can only help with the traffic flow, right ? What happens after that a road joining B and C is constructed ?

The travel time is

B-C 5 minutes.

An increasing sequence of integers is said to be alternating if it starts with an odd term, the second term is even, the third term is odd, the fourth is even, and so on. The empty sequence (with no term at all!) is considered to be alternating.

Let A(n) denote the number of alternating sequences which only involve integers from the set {1, 2, . . . , n}. Show that A(1) = 2 and A(2) = 3. Find the value of A(10).

NB I originally asked for A(20).

f is a function that maps a positive integer to a positive integer, and satisfy the properties f(n+1) > f(n) and f(f(n)) = 3n for every integer n.

a) what is f(2014) ?

b) Is f uniquely defined ? If so, what is its expression ?

Not many of you will be aware of my proud collection of racing snails. I have 8 of the speedy little critters. So fast are they, that I have an elaborate camera set up for those inevitable photo finishes.

Bonus point for identifying the title quote!

In how many ways, counting ties, can my 8 Racing Snails cross the finishing line? (For example, A and B, can finish in three ways: A wins, B wins, A and B tie.)

Tags:

Maths
Here is a question that I recently uncovered. I didn’t find the answer with the question though. For big bonus points, and a gold star – When was the question written?

The Intel clock-doubled 486DX2-66 CPU chip operates by executing a certain fraction x of instructions totally on chip at a doubled rate (66 MHz), while the remaining 1-x are executed at the normal rate (33 MHz). It’s observed that the 486DX2-66 is 76% faster than the 486DX-33 (which executes all instructions at 33 MHz). Given this and making some reasonable assumptions, estimate how much faster a clock-tripled 486DX3-99 (on chip 99 MHz; off chip 33 MHz) is than the 486DX2-66.

Tags:

Maths
As an example, 64 is 2^6. Interestingly, take away the first number (6) and you are left with 4, which is 2^2 – So…find all the powers of 2 such that, after deleting the first digit, another power of 2 remains. Base 10, no leading zeroes etc.

Tags:

Maths
Two of my students, Dumber and Dumberer play a game based on the total roll of two standard dice. Dumber posits that a 12 will be rolled first. Dumberer says that two consecutive 7s will be rolled first.

They keep rolling until one of them wins. What is the probability that Dumber will win?

Tags:

Probabilities
Show that no member of the infinite series:

10001, 100010001, 1000100010001, 10001000100010001, … is prime.

Warning: this one might be hard. e.g. the 18th term’s smallest prime factor is 722817036322379041