I have a hexagon with six equal sides, however it is not regular as two of its angles differ from the other four. Its height is 20 and its area is 1000.
What is the overall width, from point to point, of the hexagon?
Puzzle of the Week #146 - Tetrahedral Ants
Four ants are positioned at the four corners of a tetrahedron (triangular-based pyramid). At once they all move along one of the edges to another 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 #145 - Five Points on a Circle
The answer to last week's puzzle, which asked what was the probability, given three random points on a circle, that the resulting triangle would contain the centre of the circle, was 1/4.
An analogous question, asking what the probability would be of a quadrilateral whose vertices were four random points on a circle containing the circle's centre, would give an answer of 1/2.
Given six random points on a circle, the probability of the resulting hexagon containing the circle's centre is 13/16.
What is the probability that, given five random points on a circle, that the resulting pentagon would contain the centre of the circle?
Puzzle of the Week #144 - Three Points on a Circle
Three points on a circle are chosen at random, and a triangle is formed from those three points. What is the probability that the centre of the circle is inside the triangle?
Puzzle of the Week #143 - Box Packing
How many 7 x 5 x 3 blocks can you fit inside a 21 x 21 x 21 box?
Puzzle of the Week #142 - Metamorphosis
This is a fun game you can play by yourself or with others if you’re bored or stuck in a queue. You start by thinking of two four-letter-words with no letters common to both words, for example HYPE and FROG. Then you find three intermediate words, each of which changes one letter of the previous word and (if necessary) rearranges the letters to form a valid English word. Because you only have three intermediate words, each letter that you introduce to replace an existing letter must be one from the target word.
So, for example, HYPE could change to HOPE, then HERO, then on to GORE then finally to FROG.
I have yet to find a pair of words for which this is impossible. I thought by using words that had unusual letters in, the task might be made harder, but even going from NEXT to QUIZ is possible without resorting to obscure uncommon words.
So can you go from NEXT to QUIZ using only common English words?
NEXT ( ) ( ) ( ) QUIZ
Puzzle of the Week #141 - Equal Areas
Never one to miss an opportunity to recycle, this week’s puzzle uses the same diagram as last week.
Prove that the combined areas of region C and region K are equal to the area of region N.
Hint - you don’t need to calculate the area of any region.
Puzzle of the Week #140 - Rational Area
The diagram shows 6 identical circles, arranged such that the centre of each lies on the circumference of others, in a right angled grid pattern. This results in 23 regions (A to W). Assuming that the radius of the circles is a rational number, the area of each of the regions will be an irrational number, being dependent not just on the radius, but also on pi.
How can you combine a number of contiguous regions to form a combined area that is rational?
Puzzle of the Week #139 - Strange Sequence
I have a curious sequence: it begins: 2, 2, 2, 7, and then every subsequent number is the magnitude of the difference between the sum of the two previous numbers and the sum of the two numbers before that. Or in more mathematical language:
So, for instance the 5th number is 5, since (7+2)-(2+2) is 5. Remember since we are always looking for the magnitude of the difference, it cannot be negative.
What is the 123456789th number in the sequence?
Puzzle of the Week #138 - The Leaky Tank
There is a glycol tank which, to begin with, contains a mixture of glycol and water in roughly equal amounts (give or take 10%). In addition to being very imprecise about the glycol percentage, I also don’t know for certain how much liquid in total is in the tank, but it is several hundred litres.
The contents of the tank are leaking out at a rate of 2 millilitres a second. At the same time pure water is leaking into the tank at a rate of 1 millilitre per second, and mixing instantaneously with the tank contents.
At any point in time there is, say, ‘g’ millilitres of glycol and ‘w’ millilitres of water in the tank. At the point when ‘w minus g’ is at a maximum, what is the glycol concentration?
Puzzle of the Week #137 - Spare Word
Which of these is left behind when the other words are paired up in some way?
HEAR PEST PLUM REIN SLED TIER VENT
Puzzle of the Week #136 - Packing 31 Shapes in the Space of 25
This would work well as a physical puzzle, if someone wants to have a go at making it.
You have a number of identical objects whose edges are two equal circular arcs. Since each of the shaped is 12cm long and 4cm wide, 25 of them completely fill a 12 x 100 tray as shown.
However it is possible to rearrange them and fit in extra shapes. The very surprising fact is that you can fit SIX extra shapes in. Obviously none of the shapes can overlap each other, or the edges of the box. And you cannot cut the shapes up into smaller pieces.
How is this possible!?
Puzzle of the Week #135 - Odd One Out
Which of these is the odd one out and why?
Puzzle of the Week #134 - Feeling the Chill
‘42 is twice 21, therefore 42 degrees Fahrenheit is twice as cold as 21 degrees Celsius.’
Clearly the logic here is deeply flawed, however there is a sense in which the SPECIFIC conclusion is approximately true, what is it?
Puzzle of the Week #133 - Power Play
123 raised to the power of 987654321 is a number of over 2 billion digits. Please write it out in full.
Only joking! Just give me the final three digits.
Puzzle of the Week #132 - Rows and Columns
Rearrange the rows and columns to find out what this short passage reads:
Puzzle of the Week #131 - Euro Counting
If you wrote the numbers 1 to 20 in English, French and German, which letters of the alphabet would not be used at all?
Puzzle of the Week #130 - Triples
Arrange these 15 ‘triples’ into 5 related words of 9 letters each:
BUR, CAM, CES, EMB, EMM, ENT, ERT, GER, HAL, LEI, LIM, ORT, ROQ, TER, UEF.
Puzzle of the Week #129 - Power Base
What is 5 to the power of 55, minus 5, expressed in base 5?
Puzzle of the Week #128 - Five Cuboids in a Sphere
Fit five cuboids into a sphere of radius 1.
What is the greatest combined volume you can achieve?
Similarly to the Three Rectangles in a Circle puzzle a couple of weeks ago, you’ll find that a great deal of symmetry will help to maximize the volume. Again of course, the answer is not an integer, so I'll be happy to accept an answer to one decimal place.
Good luck!