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!
Puzzle of the Week #127 - Odd One Out
Which of these words is the odd one out?
APPLY GRANGE PARROT PEAK PLUG REACH
Notoriously, with odd-one-out questions, you will probably be able to find a reason why each of the words in turn should be the odd one. However, the true solution will be both convincing and satisfying, with even my choice of the odd word serving to support your hypothesis. Good luck!
Free Gift! More Binary Determination Puzzles
Here’s a little gift for you: an excel version of my Binary Determination puzzle.
Just stick it on your desktop and have a play if you ever get bored, or play it first thing in the morning to wake your brain up.
It will randomly generate one of a possible 100 billion billion billion puzzles (which is a lot more than the age of the universe in seconds). When you have completed a row or column the target number will turn green.
I’ve included macros to clear the board (but leave the target numbers as they are), or to begin a completely new game. You will need to enable macros and iterative calculation (circular references). I’ve protected the worksheet to prevent you from inadvertently changing cells that don’t need to be changed, but it’s not password protected so if you want to take a peek under the hood you are perfectly welcome!
You can download the .xlsm file from my Dropbox by following this link:
https://www.dropbox.com/s/qm9uzju27mritcx/binary%20determination.xlsm?dl=0
Enjoy!
Puzzle of the Week #126 - Three Rectangles in a Circle
Fit three rectangles into a circle of radius 1.
What is the greatest combined area you can achieve?
(I usually like to have the solution be a whole number, but this is not possible in this case. It is however possible to express the solution in the form of √a + b, where a and b are whole numbers, but I’m also happy to accept an approximate answer to a couple of decimal places).
Puzzle of the Week #125 - Right Time
I happen to look at the clock and notice that the angle between the hour hand and the minute hand is precisely 90 degrees.
How long, to the nearest second, until the angle between the hands is again 90 degrees?
Bonus Puzzle of the Week #124a - Binary Determination
I’ve introduced a couple of refinements to my Binary Determination puzzle: firstly roughly equal amounts of ones and zeros (previously there were more ones at a ratio of about 2:1); secondly a more concise way of giving the solution
Place a 0 or 1 in each of the empty cells so that in each row and column a pair of 5-digit binary numbers can be read (therefore with decimal values between 0 and 31), such that the product of the two numbers in a particular row or column is shown at the end of that row or column.
Here is an example (using only 3-digit binary numbers), so for instance in the first column, 010 (2) multiplied by 011 (3) is equal to 6, (as there is a 6 at the foot of the first column) and similarly for all of the other columns and rows. Finally, read off the diagonal of each quarter of the grid and convert back into decimal to give the solution (3,3,2,3):
Here is the puzzle:
Puzzle of the Week #124 - Binary Determination
Here’s a brand new type of puzzle that I’ve just invented this week.
Place a 0 or 1 in each of the empty cells so that in each row and column a pair of 5-digit binary numbers can be read (therefore with decimal values between 0 and 31), such that the product of the two numbers in a particular row or column is shown at the end of that row or column.
Here is an example (using only 3-digit binary numbers), so for instance in the first column, 010 (2) multiplied by 011 (3) is equal to 6 (as that is the number at the foot of the first column), and similarly for all of the other columns and rows.
Here is the puzzle:
