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In case you weren't aware: this puzzle blog has a Facebook page:
https://www.facebook.com/elliottlinepuzzles/
Each Friday morning I post the link to the Puzzle of the Week to the Facebook page, and invite people to send me their answers as a message (to avoid spoilers).
Everyone is welcome to follow my page!
In this game, the object is to make it so that all the coins are facing heads up, but on each move you must flip three coins together in a line, horizontally or vertically. For instance you could flip the first three coins on the top row, or the last three coins in the second column, or the middle three coins in the fourth row.
As it stands, the game cannot be solved, however if you flip one individual coin from the middle nine before you start, then the game can be solved.
Which of those middle nine coins should you opt to flip?
A President discovered on a TV show that you can generate an infinity of numbers even if you only had a finite number of prime numbers, and so decided that prime numbers beyond a certain size could be dispensed with.
He decreed that only the first ten prime numbers (those up to and including 29) could remain, and any numbers that contained prime factors higher than those would simply be skipped over when counting. So if you had 30 of something, and someone gave you one more, you would now be said to have 32 under this new system.
I recently did some freelance work in that country but when it came time to be paid in gold coins I was dismayed to discover that I only received half the gold coins I had been expecting! The amount I had invoiced them for was a number they didn't recognise, and so I was paid a gold coin for each of the numbers up to that number that they did recognise.
How many gold coins did I actually receive?
I have a thin strip of paper 1 inch wide. I lay it out horizontally in front of me. Along the top edge I mark two points, at five inches and ten inches from the top left corner.
I fold the top left corner down and slightly to the right, such that the fold line passes through the bottom left corner of the strip and the ten inch mark on the top edge.
I then draw a straight line from the bottom left corner of the strip, passing through the new position of the five inch mark, and continuing on its shallow diagonal trajectory until it hits the top edge of the strip.
How far along is the point where the line meets the top edge?
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 1 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.
Below is the final standings of a World Cup first round group.
The total number of goals was 16
What were the results in each of the 6 matches played?
You have two unmarked jugs, one with a capacity of 700ml and another with capacity of 1100ml. There is also a sink with a tap, however the 1100ml jug is too large to fill up at the sink. How can you measure out exactly 500ml?
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?
45 ohms
96 ohms
105 ohms
140 ohms
144 ohms
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 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