Go on and give it a try before reading on, as there are some tiny hints ahead.

Typically, a satisfying cryptarithm, or alphametic, will also ‘make sense’. For instance, the most famous, and for me the most memorable, is:

I tried to emulate this as far as possible, though it’s hard to come up with something quite as satisfying. The solution to “Chuck+Changed=Deleted”, when read one way, is my favourite. To design the puzzle: I fixed the letters to use, randomly assigned some numbers to them, and wrote a Python program to run through a decently conservative word-list checking for sums and multiplications that worked. I did this a few times, looking through for a good set of common words, and then started whittling it down to give me a set of puzzles which, when combined, gave a unique solution. The shortlist was further culled for other reasons: one was slightly more morbid and shorter than I wanted [1]; other words were pretty unclueable [2] using the brief, straightforward, single word clues I’ve gone for (apart from the more elliptic “Past-botherer” clue.)

There’s plenty more scope for other puzzles in this same style, in contast to my previous Puzzlebomb submission “In Their Prime“, which was pretty much a one off-puzzle. There’s also plenty more scope for clever individual cryptarithms that make internal sense than are available to traditional cryptarithms. This is because each individual sum no longer requires a unique solution, and without this restriction, there’s simply more of them. However you also need to find more such word sums.

I’m not sure what inspired this puzzle (possibly something at the 2014 MathsJam conference?). Whereas “In Their Prime” was a deliberate attempt to use the idea of studying inheritance of hereditary diseases in family trees, this idea somehow popped into my head fully formed. It’s certainly possible something similar may have been done elsewhere before, but I haven’t knowingly seen it.

Obviously, the English words and clues mean this puzzle isn’t as universal as “In their prime” (which did get translated onto a Spanish puzzle site at one point). I realise some people dislike wordier puzzles or find them harder, but I hope the fact that you can go back and forth between the sums and the clues (with a reduced set of letters) to work things out, makes it a bit more palatable to these folk. In fact, like the SEND + MORE = MONEY puzzle, you can deduce at least a few numbers without using any of the word clues; or you can use crossword clue logic to make some educated guesses about the word endings [3] And there is one additional hidden cryptic meta-hint lurking about if you need it.

Many thanks to the Puzzlebomb team for all their prettification and editorial contributions, and to several MathsJam guinea-pigs who I originally presented this puzzle to without any instructions (and solved it).

If you want some traditional cryptarithms, or to learn a bit more about them, Jorge Soares‘ site seems like a good place to start looking.

1. IN + END = DIE

2. DEIST

3. For instance in the aforementioned “Chuck + Changed = Deleted”.

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1. Some friends are coming over for a steak dinner. You want to cook three steaks as quickly as possible, but your grill pan only holds two at a time. Each steak must be cooked for five minutes on each side. What is the fastest you can have all three ready? Show that you can’t do any better.

Alas, when this precise situation once arose with my family at home (who says puzzles are never useful?), I was unable to convince my father not to cook the steaks in the obvious order. But at least that meant I got my steak sooner.

2. I give you two sand timers, one that measures four minutes, the other seven. Time nine minutes when I say “now”… Now!

Next, your mission, should you choose to accept it…

3. I give you two fuses that each burn take one hour to burn through from one end to the other, though not at a uniform rate. Given some matches, can you time fifteen minutes, starting whenever you want?

This problem tortured our tutor throughout a university maths summer garden party:

4. Four explorers come to a narrow rope bridge which they judge can only hold two of them at once. It’s night and they only have one torch between them, which must always be used when crossing to avoid certain death. The most foolhardy explorer can cross in one minute, the next in two minutes, the cautious one in five, and finally the limping explorer who was injured by a diabolical trap takes eight minutes. When any pair traverse together the bridge together, they must move at the slower’s pace. What is the soonest they can all safely get across to the other side?

and finally…

5. A boy, a girl and a dog are trying to get back to their home ten miles away before their dinner gets cold. The boy and girl walk at 2 miles per hour, while the dog trots along at 4mph. But they do have a single skateboard that at most one of them can use at a time: the boy and girl each riding along at 12mph, but the skilful dog can propel itself on the skateboard at an impressive 16mph. With careful planning, what is the earliest they can arrive back home?

Solutions and references some other time. If you’ve done all the puzzles (possibly before), try to generalise some!

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“I see ‘Nobel week’ as an opportunity for mathematicians to go in search of the mathematics behind each prize, rather than to retreat and complain about the lack of a prize specifically for mathematics”,

I was surprised that none of the mathsy types in my tiny corner of internet seemed to have noticed that a mathematician won a Nobel prize essentially for mathematics. After growing slightly impatient, I realised I only had myself to blame for not acting earlier, so I sketched a quick news story contribution over at the Aperiodical (it’s short and so reproduced here in full):

There may be no Nobel in mathematics, but that needn’t stop mathematicians winning one: Lloyd Shapley has just won the Nobel prize for economics, *for the theory of stable allocations and the practice of market design.*^{1}

Lloyd Shapley described himself in an Associated Press interview:

“I consider myself a mathematician and the award is for economics. I never, never in my life took a course in economics.”

But if you don’t take his word for it, look on over at his entry on the Mathematics Genealogy Project, and you’ll find his thesis is on “Additive and Non-Additive Set Functions”.

The Nobel prize website has some details on the theory of stable allocations and market design, but an old AMS feature column gives a gentler mathematical introduction, via the elegant graph theory of Hall’s Marriage theorem.

- Though technically it’s not a Nobel prize, and actually the
*Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel*. Perhaps Alfred Nobel’s made-up wife also had an affair with a fantastical economist.

If I had more time, and wasn’t trying to catch the end of a news cycle, I may have also reminded people that co-author David Gale would also have won this noblest of prizes, had he not passed away four years ago. David was described in his obituary as:

‘professor emeritus of mathematics at the University of California, Berkeley, and a puzzle lover who made fundamental contributions to economics and game theory’.

Also on the Mathematical genealogy project, Gale’s wrote a thesis that was entitled “Solutions of Finite Two-Person Games”; he invented the classic, chocolatey game of “Chomp“; wrote a maths and puzzle column, collected into the book “Tracking the Automatic Ant and Other Mathematical Explorations”; and set up MathSite, which contains interactive demonstrations of mathematics, including one of his and Shapley’s algorithm.

And finally, I *really* should have directed people to David Gale and Lloyd Shapley’s original, elegantly simple paper in The American Mathematical Monthly, which concludes:

“What, then to raise the old question, is mathematics? The answer, it appears, is that any argument which is carried out with sufficient precision is mathematical, and the reason that your friends and ours cannot understand mathematics is not because they have no head for figures, but because they are unable to achieve the degree of concentration required to follow a moderately involved sequence of inferences. This observation will hardly be news to those engaged in the teaching of mathematics, but it may not be so readily accepted by people outside of the profession. For them the foregoing may serve as a useful illustration.”

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If your answer to these questions is “No”, then please turn to page 13091204281 of the internet to have a go at the July Puzzlebomb.

If the answer is “Yes”, you can check the July solutions, including the numeric solution to *In their Prime*. But how did you find the solution? You may have successfully used trial and error, but that’s not usually very enlightening; you may have programmed a computer to do the dirty work for you (the puzzle was hand designed, but I did check the answer was unique with a bit of code). As a responsible puzzle-setter, I came up with the following possible proof of the solution. I’d be interested to hear from anyone who had a different method.

You’ve memorised the rules, right? Or you’ve got Puzzlebomb printed out or handily in another tab?

1) If you follow a gene through the family tree, each must have a different pairing in the three generations, so each prime needs three pairings with products under 120, so we certainly need the product with and to be less than the maximum, that is, , or .

2) Again, studying the diagram, you can see that no two prime genes can have same three partners. That means that at most one of 19 and 23, which only work with 2,3 and 5, can be used in the puzzle. As 19 has already been placed in the puzzle, you can eliminate 23, leaving the eight primes from 2 to 19.

3) If two of lower four {2,3,5,7} are paired together, then two of higher four {11,13,17,19} must also be paired, and their product would exceed 120. So a lower prime must always go with a higher prime. A times table is pretty necessary from now on.

4) Since 7 can never be paired with 19, so, looking up a generation you can eliminate the mid-right female (the sister of the marked mid-left male) having that gene, otherwise one of their parents would have as a check-out age. Similarly, looking a generation down you can eliminate the mid-left female (wife of the mid-left male) otherwise a child would have this disallowed level of superannuation (above 120). So the mid-right male must have the ‘7 gene’. His wife has higher death age, and this would be impossible unless 7 is paired with 11. His wife’s age involving either the 13 or 17 gene, and glancing again at the table, the only higher option is 85.

5) By similar reasoning, the mid-left pair must be 38 and 39.

6) It’s now straightforward to fill out the other generations: the males below inherit the lowest and second highest of their parents’ prime genes, while the same is true of the top generation: the grandfathers each share the lowest and second highest of the parents’ genes.

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If you haven’t already read them, there are two posts on the Aperiodical by Paul Taylor explaining the computer creation of and maths behind two puzzles that have also featured in Puzzlebomb:

- The extremely unique fractalphile “Hilbert’s Space-Filling Crossword” (only one non-trivial such puzzle exists).
- The more abundant “Spelling Bees” which have appeared several times in past issues (May, June and the aforementioned July issue), involve finding Hamiltonian paths that spell out a pair of words or phrases.

Also in the May issue, I especially liked “Word Split” a pentomino-based colouring word search (but I cheated by not breaking out the crayons). All issues can be found on the Puzzlebomb section of the Aperiodical.

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Dear Mathematician,

My partner and I are trying to buy a house. We both work in different places, and neither of us enjoys commuting. How could we decide where to live?

Fictionally yours,

Norman Mettrick

Norman,

Thank you for your intriguing and entirely imaginary letter. The short and not terribly useful answer would be…

Want to know? Read the rest of it there.

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These five sets (each of which have their elements written in numerical order) are *Schreier sets*:

The following six sets are *not* Schreier sets:

,

Can you spot the simple rule that defines whether a set of numbers is a Schreier set or not?

**Historical Background**

Jozef Schreier was born 1908/1909 in Lwów (then part of the Austro-Hungarian empire, now in Ukraine) and later based at the university there (whilst it was part of Poland). Schreier was killed in April 1943 during the Second World War in his home town of Drohobycz. In his book “Adventures of a Mathematician“, Stanislaw Ulam wrote about Schreier, his friend and collaborator:

We would meet almost every day, occasionally at the coffee house but more often at my house. His home was in Drohobycz, a little town and petroleum center south of Lwów. What a variety of problems and methods we discussed together! Our work, while still inspired by the methods then current in Lwów, branched into new fields: groups of topological transformations, groups of permutations, pure set theory, general algebra. I believe that some of our papers were among the first to show applications to a wider class of mathematical objects of modern set theoretical methods combined with a more algebraic point of view. We started work on the theory of groupoids, as we called them, or semi-groups, as they are called now.

Schreier was also a major contributor (ten questions) to the now famous Scottish book (an important book of questions, famous at least amongst Banach space theorists, and which you can read in English); in fact, he and Ulam pair were the only undergraduate student participants in the Scottish Cafe, a coffee shop where mathematicians from Lwów met, and scribbled maths directly onto the marble tabletops. The Scottish book was a notebook to save the work from being cleaned away at the end of the day.

The Baire-Schreier-Ulam and Schreier-Ulam theorems both bear Jozef Schreier’s name. He has on occasion been referred to as “Julian Schreier” (possibly by conflating him with Juliusz Schauder?): whereas Ulam’s account, the Portugese Wikipedia article, which seems to have been written by his family, and this journal from the time all name him Jozef (or a variant: Josef/Józef/Joseph). He definitely shouldn’t be confused with the contemporary mathematician Otto Schreier, who specialised in groups.

One of Jozef Schreier’s conjectures is now referenced in computer science literature: inspired by an article of Lewis Carroll, in the 1932 paper “On tournament elimination systems“, he conjectured that to find the second largest number in an unordered list requires at least comparisons, which was later confirmed in the 1960s.

I’ve included a list of some of Schreier’s publications at the end of this article.

**Schreier sets**

In 1930, Schreier constructed a counterexample to a question of Banach and Saks using the following idea:

A (non-empty) subset of the natural numbers is a *Schreier set* if its size is not larger than its least element.

The collection of all such subsets of is known as the (first) *Schreier family*. For example: and are all Schreier sets. This definition is the first my PhD supervisor showed me in my very first supervision.

You can think of building Schreier sets by picking the smallest element, say, and then filling the set up with at most strictly greater distinct numbers.

The Schreier family has the following simple properties (that make it useful when defining norms on Banach spaces, especially the Schreier space):

- The singleton is Schreier set for all
- The family is
*hereditary*: each (non-empty) subset of a Schreier set is itself a Schreier set. - The Schreier family is
*spreading*: if is a Schreier set and for all , then is also a Schreier set. (These don’t necessarily need to be in ascending order).

**Schreier and Fibonacci**

Many of areas of maths have links to the Fibonacci sequence: here’s one, involving the Schreier sets, that I haven’t seen before.

Define to be all those Schreier sets whose greatest element is . For instance . Then the number of elements the n-th Fibonacci number.

**This isn’t immediately obvious (I never noticed during my PhD, where I used Schreier sets heavily!), but here’s how to prove it. Morally, if the claim is true, there should be some way of constructing if we know both and . Let’s define two maps:**

- Replace: takes the greatest element of a set and replaces it with .
- Shift: adds 1 to each element of the set , and then appends .

For instance, , and .

I now claim that if the set is in , then is in : it will be admissible with maximum element . The new maximum is easy to see, and the number of elements stays the same, whilst the minimum element doesn’t decrease (it stays the same and only increases when acting on a singleton set). Furthermore, the mapping is one-to-one (or injective) on : if you apply to two different inputs from , you get two different outputs. And if you start with an input from , then cannot contain .

In a similar way, is a one-to-one mapping from to , and the image always contains . The number of elements increases by one, but so does the minimum element.

It’s also possible to reverse these mappings, and is an injective map to , from those elements of that contain , and is an injective mapping to from those elements of that do not contain .

Therefore, both of these maps are bijections onto their images, and:

as those sets which do contain and those that don’t, together account for the whole of . Therefore:

.

The final step of the proof is to note that and , so and .

**Ordering**

I’ve ordered the Schreier sets by writing them in binary and using the natural ordering. For example, write as 1110, which is greater than or 1010. This means we can write the Schreier sets in order as:

.

This order is just what we get when we write , or

.

In binary this looks like:

Converting to natural numbers gives:

.

Every th term in this sequence is a power of two: . You can prove this by thinking about the sum of the first Fibonacci terms and using

The differences between these are quite infuriatingly:

.

**A challenge:** if you can see a nice algorithmic way to generate one Schreier set from the previous one (not necessarily with the ordering given here), I’m sure someone would find it interesting. I would, at least. It would also be acceptable to move through each starting with , or restricting yourself to the *maximal Schreier sets*…

**Maximal Schreier sets**

During my research, I was mainly interested in the maximal Schreier sets: ones to which no further numbers could be added. With the above method of generating Schreier sets (with replacement or shifting), if you start with the maximal Schreier set and apply these two moves, you will generate only maximal subsets, and all the maximal subsets, except the singleton (The proof of this goes like the Fibonacci proof above).

**Further Afield**

I referred to the Schreier family as the *first* Schreier family. This is because higher-order Schreier families have become a focus of study. The second Schreier family is defined by choosing a Schreier set with minimum , and then adding at most Schreier sets (with higher minima), and taking the union.

For instance is a second-order Schreier set. Higher Schreier sets are then defined by induction, and transfinite induction to give infinite versions of the Schreier sets.

These higher-order Schreier families are used to define nasty Banach spaces, and often applied in conjunction with Ramsey theory. They were independently discovered by Ramsey theorists, and seem to be intrinsically linked to hereditary families of sets.

**Some Publications**

Here you will find a list of fifteen of Jozef Schreier’s papers. You can find links to many of his papers here and here. Four of Schreier’s papers are referenced on Google Scholar, and more usefully you can see recent papers that cite them. Eight of his collaborations with Ulam should be found in the book: “Stanislaw Ulam: Sets, Numbers, and Universes”, MIT Press, 1974:

“Ein Gegenbeispiel zur theorie der schwachen konvergenz”

J Schreier – Studia Mathematica 2 (1930) 58–62.

“Sur une propriete de la mesure de M. Lebesgue”

J Schreier, S Ulam – Comptes Rendus de L’Academie des Sciences 192 (1931) 539–542.

*“O systemach eliminacji w turniejach*” (“On elimination systems in tournaments”/”On tournament elimination systems”).

Mathesis Polska 7 (1932) 154–160.

“Sur le groupe des permutations de la suite des nombres naturels”

J Schreier, S Ulam – Comptes Rendus de L’Academie des Sciences 197 (1933) 737–738.

“Sur les transformations continues des spheres euclidiennes”

J Schreier, S Ulam – Comptes Rendus de L’Academie des Sciences 197 (1933) 967–968.

“Über die Permutationsgruppe der natürlichen Zahlenfolge”

J Schreier, S Ulam – Studia Mathematica 4 (1933) 134–141.

“Eine Bemerkung zum starken Gesetz der großen Zahlen”

ZW Birnbaum, J Schreier – Studia Mathematica 4 (1933) 85–89.

“Über die Drehungsgruppe im Hilbertschen Raum”

J Schreier, Studia Mathematica 5 (1934) 107–110.

“Über topologische Abbildungen der euklidischen Sphären”

J Schreier, S Ulam – Fundamental Mathematical 23 (1934) 102–118.

“Eine Bemerkung über die Gruppe der topologischen Abbildungen der Kreislinie auf sich selbst”

J Schreier, S Ulam – Studia Mathametica 5 (1934) 155–159.

“Eine Bemerkung über Erzeugende in kompakten Gruppen”

J Schreier – Fundamenta Mathematicae 25 (1935) 198–199.

“Sur le nombre des générateurs d’un groupe topologique compact et connexe”

J Schreier, S Ulam – Fundamenta Mathematicae 24 (1935) 302–304.

“Über die Automorphismen der Permutationsgruppe der natürlichen Zahlenfolge”

J Schreier, S Ulam – Fundamenta Mathematicae 28 (1937) 258–260.

“Über Abbildungen einer abstrakten Menge auf ihre Teilmengen”

J Schreier – Fundamenta Mathematicae 28 (1937) 261–264.

“Eine Eigenschaft abstrakter Mengen”

J Schreier – Studia Mathematica 7 (1938) 155–156.

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Our object of study are towers of good old-fashioned building blocks. Each block has a number written on its side, so each tower built from these blocks gives a sequences of numbers . These don’t have to be positive natural numbers, but you won’t lose much by pretending, in this post, that they are. There are innumerably many different brands of towers, but we’ll concentrate on one particular brand: the ‘Gowers Towers’. Let’s say the number written on each block represents how heavy the block is, and is inversely proportional to the length of the block. So we’d represent the sequence with the Gowers Tower pictured.

It’s worth mentioning that the Gowers Towers include every individual tower of finite height that you can build with your unlimited set of Gowers branded building blocks (and lots of infinite height, but you don’t really need to worry about those here).

Let’s pretend we’ve got a measure of the instability of a tower (the norm of the sequence), and whenever we increase the instability beyond a certain threshold, , the tower collapses.

Blocks with higher numbers are heavier, as well as narrower and perhaps inherently more unstable. How the blocks of different weights at different heights affect the stability of the Towers of Gowers is extremely complicated. However, the towers do have some nice, intuitive properties.

All the other different brands of towers, many using other brands of bricks, might not conform to the vaguely intuitive ‘physics’ I’ve set up here. However, all of them, including the Gowers Towers, have the following three properties: the zero tower made up with only zero blocks, is the most stable, having an instability of zero; if we double each number on the blocks (making them heavier and thinner), we double the instability; and if we add the numbers on one tower to those on another tower, and build a third tower, then its stability is less than the sum of the stabilities of the two original towers (which, unfortunately, is not what happens when just you slide two towers together, unless we fudge the lengths of each block).

The next property, held by the Gowers Towers, and also many nice brands of tower, is that if you replace any block with a heavier block, then a tower becomes more unstable (and visa-versa). ^{1}

**Lowering and emulating
**

The first property to keep in mind, I’ll call the *lowering* property, is that if you remove any blocks from a tower (in a sleek and skillful way, of course) the tower will drop down and be at least equally stable as before or more stable. Many other brands of tower that are naturally occurring in the wild also have this lowering property. ^{2}

The second property of the Gowers Towers is the most remarkable, and was part of Tim Gowers’ body of work for which he won his fields medal in 1998. Let us say that, in general, a brand of tower is *emulatable* if, for every possible tower, we can build an emulation: another tower with some *building restriction*, which has a similar stability. Two simple examples of building restrictions are: every tower must start with a foundation of two zero blocks; or each block must be repeated twice in a row. By similar stability, we mean that there is some finite constant such that the stability of each emulation is between and times the stability of the original. ^{3}

The Gowers Towers are *not* emulatable. This is much more surprising if you know just how very emulatable other brands of towers are. To put it another way, the Gowers Towers were a market-leader for tower brands having ‘very little symmetry’. To give you some idea of its innovativeness, the question of whether or not such a thing existed was an open for over 60 years: Gowers’ example solved a problem that Stefan Banach had posed in 1932. ^{4}

**Raising the stakes
**

Now let’s play a game. I’m going to try to convince you that, knowing the above facts, it’s a sucker’s game.

You start by choosing a *raising* move that consists of inserting zero blocks below some of the blocks. For instance, you might choose to carefully place one zero block below what were the 7th and 6th blocks, and two new zero blocks below the 4th block. This is the opposite action to lowering. ^{5}

Then on my turn, I must choose a tower with the set international standard level of stability (a stability equal to 1), such that when you perform your already-chosen move on my specific tower, it goes above a stability of and collapses. If the tower collapses I win, otherwise you do. Unfortunately, for you I can always pick a tower that will lead to my certain victory by collapsing straight after your move. But, you say, “Ha! Let’s raise by moving the construction to somewhere less windy. Like… the moon! There’s no way you can win now.” Alas, even if we took to be a billion, I know there will be a way for me to win, and my cry of “Jenga!” will echo down the hall (also used as an exclamation when any large pile of stuff tumbles sensationally).

**Proof of concept
**

To convince you, let’s suppose you could win for every possible Gowers tower for some fixed threshold . That means that your move increases the instability of every tower by at most times (it’s actually enough to consider only towers with a stability of one: you can multiply the number on each block by 1/stability to get such a tower of unit stability, and undo this multiplication after you’ve done the other moves). We have

for all possible towers.

Suppose I then did a lowering that reversed your raising move. I’ve now got back to the original tower with the original stability. So we know that

for all possible towers.

Putting both inequalities together we have

. ^{6}

This means that using your raising move, we can emulate the tower: the stability of the new raised tower is always similar to that of the original. Here, the building restriction is that there are zeros in the same positions as you inserted in your move. But this contradicts the fabulous property of the Gowers tower! That means I can always choose some tower that will collapse when subjected to your raising.

This result is surely simpler to prove directly than it is to prove the stronger non-emulatability of the Gowers tower, but given that the first property is easy to show, and the second is well-known, it seems a shame to do lots of extra work when this simple argument will suffice. If I had wanted to get extra insight into Gowers’ construction, it may have been useful to show the lack of ‘stable raisings’ from the complicated definition. But one of the ways I hoped to make use of the construction didn’t pan out, so the only result needing it in my thesis was fairly inconsequential. ^{7}

**Technical notes about Banach spaces**

If you want to find out more about Gowers’ solution to the hyperplane and other nasty Banach sequence spaces, start with this note that Gowers wrote explaining the context. The original paper of Gowers, is “A solution to Banach’s hyperplane problem” (££). However a later paper by Gowers and Maurey puts this Banach space into wider context by building it using the same basic construction technique as the Gowers–Maurey spaces. I’ve created a tiny primer to Gowers’ hyperplane counterexample by extracting material from my thesis, which repeats the material in the post above in a more serious and standard manner.

1. Different brands of towers are different *Banach sequence spaces*. The blocks here represent the *unit vector basis*, and the property noted here is *1-unconditionality*. This is in stark contrast to the similarly constructed Gowers–Maurey spaces which fail to have any unconditional basic sequences. Don’t confuse these blocks with a *block basis*!

2. The lowering property means that generalised left shifts with respect to the unit vector basis are contractive. These generalised left shifts are a certain type of spreads. Spreads are the foundation of the various construction of Gowers and Maurey. Any space with a symmetric basis (like or ) or subsymmetric basis has (like Garling’s space) has this property, as do many other spaces with unconditional bases such as Schreier space and Tsirelson space, and the James space.

3. An emulation of a tower is a strict subspace isomorphic to the whole space.

4. Gowers’ result proved something strong than Banach’s original question which was “is every inﬁnite dimensional Banach space isomorphic to its closed hyperplanes (co-dimension one subspaces)?”.

5. Lowering was a general left shift, so raising must be a general right shift. Banach Spaces with symmetric and subsymmetric unit vector bases bases have all possible right shifts bounded; spaces such as Schreier space and Tsirelson space have some, but not all.

6. for all would imply that .

7. I showed that a James-type space based on this Gowers space, instead of , doesn’t have *block right shifts*: we can’t copy and insert terms rightwards as in the traditional James space. Operators such as are not bounded.

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So to study norms, mathematicians must define them really rigourously, using something known as *axioms*. These are the basic assumptions and definitions of mathematics. Once we’ve made these assumptions we can prove what has to follow from them.

We can think of norms as a measure of distance from the origin. If you think about it in this way, the following seem quite obvious, and appeal well to our instincts. A norm satisfies the following three axioms.

- Distances are always positive!
- If the distance from your location to the origin is zero, then you must be at the origin. Or alternatively, if two points are separate then the distance between them isn’t zero. Conversely, the distance from any point to itself is zero.
- Taking a detour is always longer than travelling in a straight line. This is the triangle inequality: the sum of the length of any two sides of a triangle is longer than the length of the third.
- Now we come to axiom four. This one is tough to describe in words. Here goes. If you walk a pace forwards and then take another in the same direction, then you will have walked twice the distance of the original pace. Also it doesn’t matter whether you take a pace forwards or backwards: they will give you the same distance.

When mathematicians want to be precise, we use symbols. The distance between points and is written as . The distance from to the origin is . We say that is a norm if whenever we pick vectors and , and a number , then the following axioms hold:

- .
- If then . And visa-versa.
- .
- .

These four conditions should match with our verbal descriptions above. You may recognise them from this blog’s exquisitely hand-drawn logo.

They were pretty trivial intuitions, once we thought of as being the distance of a point from the origin (the origin above is underlined to distinguish it from the normal , though we don’t choose a different notation because the origin behaves a lot like the number zero).**In the fourth, verbally I only said: and : if you have any way of describing that in layman’s terms that includes the case , then I’d love to hear about it.**

However, even though what we said seemed trivial, now we’ve formalised the concept in a technical mathematical language, we can:

- Prove non-obvious statements about the easy cases.
- Use the same methods to prove results for cases that don’t fit into this (slightly imaginary) motivating framework. For instance, instead of points, we can think about distances between functions like or (functions can be thought of as vectors too).
- Prove statements about all possible norms at the same time (even the ones we haven’t imagined yet).
- Write complicated statements in a precise, clear and communicable way.

**Technicalities:**

Vectors have to live somewhere: we call a suitable collection of vectors that go together a *vector space. *In the definitions above both and belong to the same vector space, call it X.

The number is a scalar, often a real number in but sometimes a complex number in , or perhaps in another field depending on the vector space you are building on.

The norm is a norm on the vector space . It is a function from to .

I’ve given too many assumptions: I didn’t need to say that or that if . But I think it’s easier to accept it as excess baggage in the definition, especially when teaching.

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**Rolling it in**

In the modern classic board game *The Settlers of Catan*, it’s very important to know, when you roll a pair of dice, the frequency with which each number occurs. Resources are given players only if they have a settlement adjacent to those tiles whose number is rolled. If you build your settlements next to a tile labelled 2 or 12, it will, on average, only be productive once every 36 rolls. Tiles labelled 6 or 8 will produce resources five times in 36 rolls. It’s so fundamental to the gameplay that the relative frequencies are visualised as dots on the pieces: sixes and eights are so important, they are marked in red (rolling a seven does something different).

**Puzzle**

The well-known puzzle is:

By relabelling the faces of two dice, can you design a new, unusual pair of six-sided dice that achieves rolls with the same frequencies as a pair of normal dice? All the faces must have a positive number of spots.

If I didn’t require you to use a positive number of spots on each face: then dice labelled {0, 1, 2, 3, 4, 5} and {2, 3, 4, 5, 6, 7} would work as a pair. If you allow negative numbers, there’s infinitely many solutions!

**Solution**

There is a unique solution, known as the Sicherman dice: label the faces {1, 2, 2, 3, 3, 4} and {1, 3, 4, 5, 6, 8}.

A good start is to observe that 2 can only be rolled as 1+1. To make 3, at least one of the two dice must have a 2 on it. After that, you must somehow account for another 3, that must arise as 1+2. Now you are faced with a choice: if you put a 2 on both dice, the only way to complete it gives you the normal dice. However, if you place both 2s on the same dice, the only way of completing is to make the Sicherman dice.

You can also use generating functions and cyclotomic polynomials to decide how to label the dice, algebraically.

**Dice and Dissection**

A nifty detail, I haven’t seen elsewhere, is that you can represent both of these pairs of dice as a different dissection of the following shape:

Each coloured horizontal bar shows the contribution from a normal die, translated right by the numbers 1 to 6 from the other die. The zig-zags are the contributions from the first of the Sicherman dice (which has an interval of natural numbers), and they are translated to the right by the numbers on the second Sicherman die: {1, 3, 4, 5, 6, 8}.

The height at which each box is placed is irrelevant, so I chose them to make the nicest possible diagram. The next picture is a different way of describing the same pair of Sicherman dice. The fact that the first diagram above happens to give nets for a cube, if you cut out one of the Sicherman regions, seems merely coincidental; it is impossible arrange a similar net for a cube from the mono-coloured regions which give one of the usual dice (can you see why?).

These pictures may be the quickest, most intuitive way to explain to friends why using your cherished pair of Sicherman dice to play Catan won’t affect the tightly balanced gameplay. Good luck with that.

**More Dissections**

You can also use these tessellations to easily spot other ways of creating this standard probability distribution. For example, you can use a tetrahedral die and a nine-sided die (not a platonic solid).

You can read off the number of spots on each face from the diagram (and subtracting one). The values are: looking at the light-blue tetronimo in the lower left, {1, 2, 2, 3}; and from the lower leftmost square of each differently coloured tetronimo, {1, 3, 3, 5, 5, 5, 7, 7, 9}. You can find other strangely-sided dice pairs with the Sicherman property in a similar way.

If we include zero and negative numbers, the six-sided translated solutions {1-n, 2-n ,3-n, 4-n, 5-n, 6-n} and {1+n, 2+n ,3+n, 4+n, 5+n, 6+n} give the same pictures as the standard dice. That reassures us that these are rather dull solutions.

**Further questions
**

Can we extend to three dice? Some puzzles in increasing order of difficulty:

- Quickly think of a set of three six-sided dice that give the same probability distribution as three standard dice (all faces still strictly greater than 0).
- One of your three dice cannot be labelled {1, 2, 2, 2, x, y}. Can you see why not?

The next two questions need some algebra:

- Prove there are no further such sets of three six-sided dice. [Hint: study Proposition 9 and consider ].
- Considering the series expansion of for show that, to emulate six-sided dice with positive faces, you must always use a combination of normal and Sicherman dice.

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