What is going on with this puzzle? Well, the question marks are probably important, and it looks as though each line is a clue. Starting at the top, the four stars in front of Turing could be "Alan". The next line is perhaps another word for unfortuantely, maybe "Alas". Hang on, that's only one letter different from Alan. Maybe this is a word ladder, where each answer is related to the one above by changing a single letter. Working on that basis we can complete all the words associated with each line. Looking at the lines above and below each set of question marks gives enough information to work out what the question marks must be.

**** Turing	ALAN
Unfortunately	ALAS
Money or food given, often as charity	ALMS
Upper limbs	ARMS
Crafty creative pursuits	ARTS
What somebody does on stage	ACTS
FR****L - a mathematically complicated geometrical figure	ACTA
????	OCTA
After Seps but before Novs	OCTS
Deeds, not words	ACTS
V*R*A*ION* - a type of calculus	AITS
Targets or goals	AIMS
A***O*ITY - strong hatred	NIMS
Slang for tasty food?	NOMS
A****LOUS - unexpected	NOMA
????	NONA
Da Vinci's Lisa?	MONA
A member of a religious order	MONK
Make fun of	MOCK
A leaf that cures nettle stings	DOCK
A floor of a ship	DECK
????	DECA
****N - type of nut	PECA
Something a chicken might do	PECK
The top of a mountain	PEAK
A type of decayed plant matter, sometimes used as fuel	PEAT
Excellent (slang)	PHAT
????	WHAT
An informal conversation	CHAT
A warm piece of clothing	COAT
A small ship	BOAT
Rhythm	BEAT
**** and tidy	NEAT
????	NEXT
Small lizard	NEWT

Taking only the words associated with the question marks yields the question: OCTA, NONA, DECA, WHAT NEXT?

Octa, nona and deca are prefixes that mean 8, 9 and 10, respectively, so we are looking for a prefix that means 11. The answer is undeca.

This code is inspired by esoteric programming languages, which each use a restricted set of control symbols. The code is arranged into different blocks consisting of a letter, an up or down arrow, a number and then a right arrow. This looks a bit like a sequence of instructions all following the same general pattern.

If the code is an algorithm, or list of instructions, then we just need to work out what the instructions mean. Each block ends with a right arrow, so that could mean "output". What about the letter/arrow/number combination? Well, the obvious thing would be to go through the alphabet from the starting letter in the direction of the arrow by the indicated number of places. For example e ↑ 18 would be w , which is 18 places above e in the alphabet.

Following these instructions reveals the question, after adding spaces and punctuation:

What is the lowest positive value of x for which cosine of x equals x in radians?"

In fact the equation $\cos x = x $ must be solved numerically using your favourite root-finding method . There is a unique solution: $x\approx 0.74$ in radians, so the answer is 0.74.

Obviously, the solution has something to do with the numbers in the boxes, but what? Is there anywhere that you might have seen numbers in boxes like this before? If you search for numbers in boxes in science, you quickly find references to the periodic table of the elements. In the standard convention the number at the top is the atomic number and the number at the bottom is the atomic mass.

Converting the boxes to the corresponding chemical symbols for the elements gives:

W	H	At	I
S	Ni	Ne	P
Lu	S	O	Ne

In other words, the question is: WHAT IS NINE PLUS ONE?

The answer is, of course, 10.

This collection of runes looks like a substitution cipher. There are quite a lot of dots. Perhaps those correspond to spaces. Working on that assumption, then using frequency analysis allows decryption of the code to give the below question, after adding suitable punctuation.

A small robot rolls on a flat plaza. It starts at an origin and rolls four units forward pivots right orthogonally and rolls six units forward. A bright lamp is hung two units up from this robots starting origin. How long is minimum path from this robots final spot to this lamp?

If we assume that the direction up is out of the plane (plaza) and choosing our origin to be where the robot starts then:

By using Pythagoras, the distance of robot from origin on plaza is $$\sqrt {4^2 + 6^2} = \sqrt{16 + 36}= \sqrt{52}$$

Now we consider another triangle which is the distance from the origin to the robot's final location on the plaza, the distance from the orgin to the map and the distance that we want: lamp to the robot's final location. Using Pythagoras again we find that the desired distance is:

Distance of lamp from robot is $$\sqrt{2^2 + (\sqrt{52})^2} = \sqrt{4 + 52} = \sqrt(56) \approx 7.48.$$

The answer is 7.48.

Each line has a different geometric shape (actually a Platonic solid), a letter in a script font, F, E or V, and then a collection of letters. What do these have to do with each other? Maybe the shape and letter tell us something about how the letters are encoded?

The letters F, E and V are commonly used in solid geometry to mean Faces, Edges and Vertices. The first shape is a tetrahedron and the letter is E. A tetrahedron has 6 edges, so let's try shifting the letters by 6 places in the alphabet: ULCABNWCLW becomes ARIGHTCIRC, which starts to look like words.

Pursing this idea, the next line starts with an octahedron and the letter is E. An octahedron has 12 edges and shifting the letter forward by 12 places in the alphabet gives: ULARCONEHA.

Carrying on shifting the letters on each line by the indicated number of places gives the following message:

A RIGHT CIRCULAR CONE HAS TWICE THE RAIDUS OF A RIGHT CIRCULAR CYLINDER. IF THE CONE AND THE CYLINDER HAVE THE SAME VOLUME, WHAT IS THE RATIO OF THE CONE'S HEIGHT TO THAT OF THE CYLINDER?

We can use the formulae for the volume of a cone and volume of a cylinder to solve this problem. Let $R$ be the radius of the cylinder. Then the volume of the cylinder is: $$ V_{cyl} =\pi R^2 H,$$ where $H$ is the height of the cylinder. The volume of the cone is: $$ V_{cone} = \pi r^2 h / 3,$$ where $r$ is the radius, and $h$ is the height of the cone. The radius of the cone is twice the size of the cylinder's radius, so $r = 2R$. If the volumes must be the same then $$ \pi R^2 H = \pi (2R)^2 h / 3 \Rightarrow H = 4 h / 3 \Rightarrow h/H = 3/4 = 0.75.$$

The answer is 0.75.

The first question is quite simple to spot, it's just written around the edge of the circle: HOW MANY RADIANS ARE THERE IN A FULL CIRCLE PLEASE?

The second question must be something to do with the numbers in the middle of the circle, but what could they represent? Is there a clue in the first question? The numbers are all in the range $0$ to $2\pi$, which is the range of possible angles in a full circle radians. They could indicate the angles of the letters around the edge, but where is zero? The last number is 6.13 and if the numbers represent a question, then the last symbol is likely to be a question mark. That suggests that maybe zero radians is at the top of the circle.

There are 42 symbols around the circle and $2\pi / 42 \approx 0.15$, so if 'H' is at zero, then the $n$-th symbol after 'H' will be at the angle $0.15 n$. In other words, the first number, 0.3, would correspond to the third letter (the second after the 'H'), which is 'W'. Carrying on in this fashion yields the second question:

WHAT IS THE RADIUS OF A CIRCLE OF CIRCUMFERENCE TEN TO TWO DP?

The answer to the first question is $2\pi$. The answer to the second question is $10/2\pi$ to two decimal places, which is $1.59$. The product of these two answers is $$ 2\pi \times 1.59 \approx 9.990,$$ to three decimal places.

The exact answer would be 10.000, but the second question specifically asks for the answer to two decimal places.

The answer is 9.990.

Alan Turing Cryptography Competition 2026 is organised by the The Department of Mathematics at The University of Manchester.
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