Puzzle of the Week #178 - Tangent Curve

The line begins in a vertical downward direction, transitions into an arc of radius 60, then into another arc with a tighter radius, which just touches the horizontal dashed line. Then there is an arc of radius 50 in a clockwise direction, then a horizontal line. At each transition between line and arc or arc and arc the tangents are continuous.

What is the missing radius?


Puzzle of the Week #177 - Skyscrapers and Empty Lots

Each of the 25 squares contains either a building of height 1, 2, 3 or 4, or an empty lot. Each row and each column contains exactly one of each of those.

The numbers in the twenty ‘vantage points’ around the edge denote how many buildings you can see looking along that particular row or column, so for instance if the 2 building was closest, followed by 1, 4 and 3 in that order, the vantage point would say 2, as you can see the 2-building and the 4-building, whereas the 1-building and 3-building are hidden behind taller buildings.

Here’s an example (with only 1, 2 and 3 height buildings) which might make it clearer, followed by the puzzle itself:

skyscrapers.JPG

Puzzle of the Week #176 - Nine Point Circle

You might be familiar with the famous ‘Nine Point Circle’ theorem, which says that for ANY triangle the 3 midpoints (shown below in red), the 3 perpendicular feet (shown in green), and the 3 points midway between the each of the vertices and the orthocentre (shown in blue), will all lie on a circle. (Of course for an equilateral or isosceles triangle some of these points will coincide). But I have discovered there is something quite special about the 9 points of a triangle with angles 45, 60 and 75.

What is so special?

9 point circle.JPG

The figure is to scale so you may be able to guess the answer.

Puzzle of the Week #173 - Tournament

There are 16 teams in the knockout stages of a football tournament. For the sake of argument, say that each team has a distinct ranking from 1 (best) to 16 (worst), and that in any individual match, the better team will always win and progress to the next round.

The first round matches are randomly arranged, and the eight winners of those matches are randomly arranged in four matches in the quarter final etc.

Which of the following scenarios is more likely:

The 2nd ranked team is knocked out before the semi-finals, or

The 7th ranked team progresses at least as far as the semi-finals?

Puzzle of the Week #170 - Football Squad

A squad of 20 players (3 goalkeepers and 17 outfield players) splits into two equal teams for a training match (with any unchosen players reduced to spectating). Each team must have exactly one goalkeeper, but otherwise the players are entirely interchangeable.

It is calculated that there are a little over 5 million distinct* ways to select the teams.

*(so, for instance, if it were three-a-side, ABC v DEF would be the same as FDE v BCA).

How many players on each team?

 

Puzzle of the Week #167 - Anagrhymes

It’s been a while since I made a word puzzle, so here is one I hope you’ll like:

anagrhymes.JPG

Via a series of steps, change from one word to another, firstly changing to an anagram of the previous word, secondly to a word that rhymes with the previous word, and subsequently alternating between the two types of changes.

For instance if you wanted to change from CHAR to CHARM you could change CHAR to ARCH (anagram), then ARCH to MARCH (rhyme), then MARCH to CHARM (anagram).

Can you change from THING to PRIEST:

Puzzle of the Week #165 - Cube Ants

Eight ants are positioned at the eight corners of a cube. At once they all move along one of the edges to another adjacent corner, each choosing at random from the three other corners available.

What is the probability that the ants will perform this manoeuvre without any of them having to pass another coming the other way along the same edge or ending up at the same corner as another ant?

Puzzle of the Week #164 - Number Sequence

I’ve discovered this interesting way of making sequences. Start off with any three numbers, then any subsequent number is the sum of the previous number and the number before that, minus the number before that.

In mathematical terms a(n) = a(n-1)+a(n-2)-a(n-3)

If you start with, say, 1, 1, 1, the sequence continues with 1s forever.

If you start with 1, 2, 3, the sequence continues to count up through all of the integers in turn.

I plugged in some random numbers as the first three terms and looked way down the list that was produced:

The 243rd number in the sequence is 1092.

The 302nd number in the sequence is 1355.

The 577th number in the sequence is 2595.

What were the first three numbers in the sequence?