Puzzle of the Week #401 - Chunkwords: Mathematical

I have taken ten 9-letter mathematical terms and split them into 3-letter chunks. Can you reassemble them?

 

AIN        ALG        AME      ANG      ANS       BLE         COT        DER        DIV         DOD

DRA       ECA        ENT        ERA        EXP        GON      ION        ISI           NSF        NUM

ORI         ORM      PAR        QUA      REM       TER         THM      TIC          TOR        TRA

Puzzle of the Week #400 - Peace Treaty

To celebrate reaching 400 Puzzles of the Week, I thought I’d bring out one of my favourite puzzles. This one first appeared in my book ‘Paddocks’ available from lulu.com or amazon.

On the island of Honeycombia live two warring tribes: the Crosslanders and the Noughtlanders. They have stopped fighting for now, but a permanent ceasefire is in your hands.

It is your job, as chief peace negotiator, to divide the island in a way that satisfies both tribes.

Each tribe wants their own territory to be in one piece, so that they can travel from any point in their own territory to any other, without crossing the other tribe's territory, and without going to sea.

Also, neither tribe wants the other to be able to build certain types of structure: a diamond mine would require four hexagons together in a diamond shape, and a car factory would require five hexagons together in an axle shape. Consequently neither shape can appear on the island, in any orientation, for either tribe.

Oddly, neither side seems to care who ends up with more territory, as long as all of the hexagons are assigned to one of the two tribes.

Both tribes have already laid claim to some areas of the island. Can you assign all of the other hexagons to one or other of the tribes, so that a lasting peace can emerge on Honeycombia?

Hint: each tribe can only have one length of coastline. If the coastline switched back and forth between the two tribes, it wouldn't be possible for one territory to be in one piece without dividing the other territory in two.

 

Puzzle of the Week #398 - Find The Radius

A circle is inscribed within a quarter circle. A line of length 1 extends from the vertical edge to the arc of the quarter circle is parallel to the horizontal edge of the quarter circle, and tangent to the inscribed circle.

What is the radius, R, of the quarter circle?

1 point for a numerical solution, 2 points for an exact solution using nested square roots, 3 points for finding a quartic polynomial whose real positive solution is R.

 

Puzzle of the Week #394 - Coin Paradox

I’m thinking about a sequence of coin flips resulting in either heads or tails, with equal probability. Which of the following statements are true?

 

In a game where I win if we flip HTH on consecutive throws and my friend wins if we flip HTT on consecutive throws:

 

1) If we just toss the coin three times in succession, we each have an equal probability of winning (although most of the time neither of us would win).

2) If we keep flipping until either HTH or HTT comes up, we will win the game with equal probability.

3) If we keep flipping until either HTH or HTT comes up, the average number of flips it will take is on average the same, whether I win or my friend does.

4) If we each have our own coin, and I keep flipping until I see HTH on consecutive flips, and my friend keeps flipping her coin until she sees HTT on consecutive flips, we will both take the same number of flips on average.

Puzzle of the Week #393 - Two Point Oh Eight

52/25 = 2.08 precisely

572/275 = 2.08 too

 

52 and 572 are the only two- and three-digit numbers which, when divided by the reversal of their digits, is exactly equal to 2.08.

 

How many 25-digit numbers are there, that when divided by their reversal become equal to 2.08?

 

This might look like an exercise in coding, but it isn’t. In fact I devised the puzzle purely on paper, and it could be solved as such too.

 

Puzzle of the Week #391 - Five Towns 2

This is a very similar puzzle to last week’s, with a couple of minor tweaks.

 

Crosstown is on a straight line between Lefton and Righton.

Crosstown is also on a (different) straight line between Upton and Downton.

Unlike before, these two lines do NOT need to cross at right angles.

As before each of the distances between towns should be a unique whole number of miles.

However now there is the added restriction that none of the distances should be a multiple of 3 (although U-D and/or L-R could be as they are each the sum of two separate distances).

To complete the puzzle you just have to find a set of distances which satisfy all of the restrictions. For extra kudos, can you get the total length of the eight roads to be less than 400 miles?

Puzzle of the Week #390 - Five Towns

The towns of Norton, Sutton, Weston, Easton and Middleton are arranged such that the Norton-Sutton road and the Weston-Easton road cross at right angles in Middleton.

The distance from any town to any other town is unique.

Each distance is an exact whole number of miles.

What is the smallest possible total area of the diamond shape formed by the outer four towns?

Puzzle of the Week #389 - Strings

I have a number that is a string of 1s, followed by a string of 2s, followed by a string of 3s. For example 111233. There must be at least one of each digit.

Similarly I have a second number that is a string of 3s, followed by a string of 2s then a string of 1s, with again at least one of each of the different digits. For example 332211111.

Adding these two numbers together I get the answer 44443444.

How many different possibilities are there for my two numbers?

Puzzle of the Week #388 - Ten Pin Bowling

This game uses all 100 scrabble tiles, including the two blanks (which can represent any letter of your choosing).

The scoring system is the same as that of real ten-pin bowling: you get points for each word, equal to how many letters in the word. In addition, if you get a spare (use all letters in one frame using two words), you get bonus points equal to the next word you score.

If you get a strike (a ten letter word), you get bonus points equal to the next TWO words you score.

If you only get one word in a frame, and it's not a strike, then for the purposes of bonus points, you get a zero length word too.

In real tenpin bowling, if you get a strike or a spare on the tenth frame, you get an eleventh frame to determine your bonus points, and if you were lucky enough to get a strike on the tenth AND eleventh frames, you would get a twelfth frame.

In this game, there are no eleventh or twelfth frame, so to determine any bonus points you are entitled to after the tenth frame, look back at the words you scored in the first couple of frames.

The 100 scrabble letters have been assigned randomly into their frames as below (which means I can play along with everybody else). Generally a score of 100 or more using relatively common words is a pretty good benchmark, although a higher score is probably possible by poring over lists of more obscure words.

 

Puzzle of the Week #387 - Five Circles

I have five circles arranged tangent to one another as below, and the triangles formed by connecting some of their centres result in three right-angled triangles.

Can you find an arrangement where the radius of each of the five circles is a whole number?

For bonus points, what is the minimum arrangement where each of the radii is a whole number (minimum sum of the five radii)?

Puzzle of the Week #385 - Odd Prime Cycle

Start with an odd prime number

Add 1, then choose an odd prime factor of this new number,

Add 2 to that number, then choose an odd prime factor of this new number,

Add 3 to that number, then choose an odd prime factor of the new number,

Add 4 to that number, then choose an odd prime factor of the new number,

etc, until the odd prime factor you get is the odd prime you started with.

 

For instance, if you started with 5:

5+1 is divisible by 3

3+2 =5

 

Other than 5, what could the starting odd prime number have been?

Puzzle of the Week #383 - Smallest Square

What is edge length of the smallest square that can contain a shape with equal (non-zero) perimeter and area?

 

Examples of such shapes are a circle of radius 2, which has both area and perimeter equal to 4*pi, or a 6,8,10 triangle, which has both area and perimeter equal to 24. My shape might be made up of straight lines, or curves, or both.