If someone in Port-au-Prince writes to someone in Zagreb, someone in Bratislava writes to someone in Bishkek, someone in Rabat writes to someone in Yerevan, and someone in Baku writes to someone in Cape Town, where does someone in Sofia send their letters?
Here is an interesting mathematical game to play. The idea is to reach a total by multiplying and/or subtracting positive integers, whilst minimising the sum of those integers. Rather than explain I’ll demonstrate:
For example if the target is 99 you could have:
11 x 3 x 3 = 99, giving a score of 17
Alternatively you could save a couple of points thus:
5 x 5 x 4 – 1 = 99, giving 15 points
But even this can be improved upon:
3 x 4 – 1 x 3 x 3 = 99, scoring 14 points.
Note that each calculation, whether multiplying or subtracting, is performed stepwise, rather than using the BODMAS order of operations.
So, the target I have chosen for the puzzle is 12345. What is the lowest score you can achieve?
I should point out that I am not confident that the answer that I have is the best achievable (twice before I thought I had the best answer and then found a better one). If your score is higher I’ll be able to tell you that a lower score is achievable, but I am fully prepared to discover that my score is not the best either!
Additionally, if anyone can find an algorithm that finds the best solution in the general case I’d love to hear of it. At present all I have been able to establish is a loose set of rules of thumb.
To combine resistors in series, you add their resistances together. For example 10 ohms and 20 ohms combined in series gives 30 ohms.
To combine resistors in parallel, you take the reciprocal of the sum of reciprocals of the individual resistors. For example 12 ohms and 24 ohms combined in parallel gives you 8 ohms (1/12 + 1/24 = 1/8).
You have 5 resistors, each with a rating of 120 ohms. Combining them in different ways it is possible to achieve a variety of resistances, for example, the arrangement below results in a 100 ohm resistance (60+40).
One of the following values is NOT possible using the 5 x 120 ohm resistors, which is it?
In a recent half marathon with a few thousand runners, the race numbers began at 1 and counted upwards in the usual way.
I happened to notice that the average sum-of-digits of the four digit race numbers was precisely the same as the average sum-of-digits of the three digit race numbers. I also happened to notice that the very highest number was an exact multiple of 17.
How many runners were there?
I have two identical maps, except that one is printed as A3 and one is A4. I rotate the A4 map 45 degrees anti-clockwise and position it over the A3 such that the top left corner of the A4 map lies on the left edge of the A3 map, and the bottom left corner of the A4 map lies on the bottom edge of the A3 map.
There will be exactly one point such that if I put a pin through the two maps, they will pinpoint the same place on the map. Where should I put the pin?
There is a regular 2018-sided polygon engraved in the Nazca Desert in Peru. The corners are alternately coloured red and green. Any pair of opposite corners are exactly 2 miles apart.
You stand on one of the green corners, and measure your straight-line distance to each of the 1009 red corners (in miles). You multiply together all of these distances.
What do you get?
(Maths hint: try smaller numbers of sides first to see if you can detect any pattern.)
I’ve combined one of my early puzzle creations, Zipline (see https://www.janko.at/Raetsel/Zipline/index.htm for examples you can play online), with an established Japanese puzzle type, Futoshiki, to create this new type of puzzle.
The numbers 1 to 16 need to be placed in the squares according to the following rules:
The symbols denote that the number in one square is greater (>) or less (<) than the number in an adjacent square.
1 and 2 appear in the same row or column (although not necessarily adjacently), 2 and 3 appear in the same diagonal, 3 and 4 appear in the same row/column, 4 and 5 in the same diagonal, etc. (So, consecutive numbers where the lower number is odd will appear in the same row or column, and consecutive numbers where the lower number is even will appear in the same diagonal).
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?
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?
The probability of the centre of the circle being within the pentagon is 11/16.
The general formula for an n-sided shape is: 1 - n/(2^(n-1))
If you're interested, the derivation of the above formula is as follows:
The puzzle asked, for five points positioned at random, what is the probability of the pentagon formed by them containing the centre of the circle.
Let’s switch is round and ask what the probability is of missing the centre. For this to be the case, all of the points must be in the same half of the circle, but that half could be 0 to 180 degrees, or 180 to 360, or 63 to 243, or any of the countless other ways. So let us pick a point, one of the five random points, and (temporarily) call that our zero, our datum. What is the chance of the other 4 points being less than 180 degrees clockwise from the datum? Simply ½ x ½ x ½ x ½, or 1/16.
Each of the five points can be taken as the datum in turn. So is it just a matter of adding together those five probabilities (or, as they are equal, simply multiplying 1/16 by 5)? Well yes, since at most one datum can result in all the other points being less than 180 clockwise from that point, the five cases are mutually exclusive, and it is indeed just a matter of adding them together.
From here it is easy to see what the general equation is. For n points, the probability of not containing the centre is:
Giving the sequence ¾, 4/8, 5/16, 6/32, 7/64, 8/128 etc (before simplification, to demonstrate that the numerator increases by one and the denominator doubles with each new term)
Then the probability of containing the centre is merely the complement of this, or 1 – n/(2^(n-1))
Resulting in the sequence ¼, ½, 11/16, 13/16, 57/64, 15/16 etc.
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?
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?
How many 7 x 5 x 3 blocks can you fit inside a 21 x 21 x 21 box?
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
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.
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?
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?
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?
Which of these is left behind when the other words are paired up in some way?
HEAR PEST PLUM REIN SLED TIER VENT
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!?