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Below is a portion of an ellipse. The semi-minor axis has length 5. A 45 degree diagonal line from the centre to the edge of the ellipse has length 7.
What is the length of the semi-major axis?
If it helps, the general equation for an ellipse is:
(x/a)^2 + (y/b)^2 = 1, where a and b are the two semi-axes.
Here’s something a little different for you this week. This is a game I invented which you can play on your own or against others. All you need is a bag of tiles from a standard Scrabble set. Randomly draw out ten tiles and arrange them in a triangle. This is the first frame of the game of ten-pin bowling. If you can form a ten-letter word, you have scored a strike. If you can form two words using each of the ten letters exactly once each, this is a spare. If you can do neither, try to use as many letters as possible in at most two words. Scoring is exactly the same as for ten-pin bowling. Continue for all ten frames (there are 100 letters in a scrabble set), and then tot up your score.
If you’re not familiar with the scoring system of ten-pin bowling I’ll briefly explain: 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 and 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 and second frames.
So that’s the general outline of the game. For the purposes of this Puzzle of the Week I’ve randomly generated a complete game. Try to resist using solving tools as it takes the fun out of it, and see what the highest score you can achieve by only spending a minute on each frame.
For reference I scored a modest 95 points. But then this is a particularly tough board! I don’t believe there are any strikes up for grabs (prove me wrong!), but a spare is possible on at least some of the frames.
In Elbonia they only have three denominations of coins: 15 ELB, 21 ELB and 35 ELB. Although each pair of these coins have a common factor, all three do not. This means that if the value is high enough, any value can be expressed exactly with no change required. But what is the highest amount that cannot be expressed exactly?
x^2 = a^2 + 4b + 1
y^2 = b^2 + 4c + 1
z^2 = c^2 + 4a + 1
a, b, c, x, y, z are all positive integers
a > b > c
x > y > z
What are the values of the six integers?
(There is a really neat algebraic trick you can use here. First see if you can calculate x in terms of a).
I have a number of coins. I throw them all, count how many heads and how many tails and multiply those two numbers together. I calculate what the expected (average) answer should be and discover it is exactly three times the number of coins. How many coins do I have?
(For instance if there were three coins, there are eight possibilities:
HHH TTT HHT HTH THH TTH THT HTT
The first two result in 3 x 0 = 0, and the other six result in 2 x 1 = 2, so the average answer is 1.5)
I have a six letter word, a clue to which is: ‘Something Winston Churchill often is’.
If I change the first letter of my word to the letter that follows it in the alphabet, and then reverse the entire word, I get a new word.
What was my word?
In the following diagram, the ‘staircase’ is made up of horizontal and vertical lines of equal length. The ends of these lines determine the radius of each of the rings, whose shared centre is the end of the final horizontal line.
If the overall shape has an area of 100, what is the area of each of the individual regions A to G?
A quotation has been hidden in the large grid below. The first word can be read off (left, right, up, down, but not diagonally), and its letters eliminated. Later words in the quotation may not be made of consecutive letters until the letters of previous words have been eliminated. All the letters in the grid will be used exactly once then eliminated.
As an example, the phrase ‘In the nick of time’ has been hidden in the example grid:
“4, 3, 6, 3, 8: 3, 8, 3, 5, 9. 3, 2, 3, 4, 5, 3, 6”
If we count a workday as a day that is not Saturday or Sunday and is not a Bank Holiday, the most you will see in a given calendar month is 23. This happens twice in 2019: July and October. In the UK there are Bank Holidays at New Year, Easter, May, August and Christmas. Is it possible for a calendar year to have no months with 23 workdays?
Four identical pieces forming a square with a four-leaf-clover shaped hole can be rearranged (after flipping half the pieces) into a rhombus with three circular holes. The shorter diagonal of the rhombus is the same as the side length of the square.
Rounded to the nearest whole number percentage, what proportion of the square is shaded?
Grace and Ruby live in a house next to a fast flowing river. One day they decide to travel a few miles down the river and then return home. Grace is on foot, jogging along the riverbank at a speed of 5mph. Ruby takes the boat, which can travel at 9mph relative to the current of the river. They both set off at the same time, turn around at the same spot and arrive home at the same time. How fast was the current of the river?
Phyllis and Dilys are identical twins. They are each independently given the same 4-digit number.
Phyllis takes the number and converts it from decimal (base 10) to base 4, and writes down the 6-digit result.
Dilys simply writes the first and last digits of the number followed by the number in its entirety.
They are astonished to find that they have both written down the same 6-digit number!
What was the original number?
In other words, which number ABCD, when converted from decimal to base 4 becomes ADABCD?
Four semicircles are arranged around the edges of a rectangle. What are their respective radii?
This is a great deal more difficult than ‘Three Semicircles in a Square’. If it helps I can tell you that there is exactly one solution, and that each of the radii are rational numbers (so if you find it useful – and I’m not at all sure you will - you can scale the whole thing up by some factor, solve as a Diophantine set of equations, and scale back down again). I myself found this very difficult to solve, so if you discover a route to the solution that is not too difficult, I would be interested in knowing it.
Three semicircles are arranged around the edges of a unit square. What are their respective radii?
A travelling salesman needs to visit each of the black squares in turn and return to where he started. He can choose to start wherever he wishes. He traces a zero-thickness path, can travel at any angle (this gridlines are only a reference) and doesn’t need to go to the middle of each black square, it is sufficient for his route to touch a corner or an edge. If each small square measures 1 x 1, what is the length of the shortest route?
Given a string with three each of three different letters ordered thus:
A A A B B B C C C
It is possible through a series of three moves to change the order into:
A B C A B C A B C
A ‘move’ consists of taking a section of the string and reversing the order of the items within it. The brackets show the section to be reversed in the subsequent move:
A(AABB)BCCC -> ABBA(ABC)CC -> AB(BACBAC)C -> ABCABCABC
Using the exact same idea, how many moves will it take to change between the following?
A A A A A B B B B B C C C C C D D D D D E E E E E
A B C D E A B C D E A B C D E A B C D E A B C D E
The convergent sum of the following infinite sequence, in its simplest terms, is a fraction with a square number on the top and a factorial number below.
1/7 + 1/16 + 1/27 + 1/40 + 1/55 + 1/72 + … + 1/n(n+6) + …
What is it?
Footnote: the partial sum converges extremely slowly, such that if you add the first million terms you will only get the first 5 decimal places. However there exists a very nice trick to allow you to work out exactly the number the infinite sum converges upon (without having to do an infinite number of calculations!).
B is the midpoint of AC
D is one third of the way along AE
CD is perpendicular to BE
AE is (√2) times the length of AC
What is the angle at A?
The towns of Abbottsville, Beresford and Christchurch all lie on one straight road. I embark on a journey from Abbottsville, through Beresford, to Christchurch, and back the same way. Each day I cover 1 mile more than the day before. It takes me 10 days to travel from Abbottsville to Beresford, 11 days to travel from Beresford to Christchurch and 12 days to travel from Christchurch (through Beresford) to Abbottsville.
What are the distances between the towns?
The open beta has now started 🎉
Also, there’s now a little trailer on the store page (link to the trailer: https://www.youtube.com/watch?v=is8n293Z_zs).
Unfortunately, Google doesn’t offer Instant Apps for beta versions, so at the moment there is no free
demo available. We can hopefully get that sorted in the next few days.
Moritz Neikes has done a tremendous job with the app, thank you Moritz!