There's a puzzling little problem I found in one of my old maths textbooks (Basics of Madthematics). In the middle of a whole bunch of normal problems on calculus and algebra and whatever, suddenly there's this. The book has no answers either, and while some of the questions are definitely tough, I've had a lot of trouble with this one. I've devised plenty of possible phrases over the years, but never found a solution that I feel reflects the whole problem, or has four different ways of constructing it from the puzzle. Thus I thought I'd bring the problem to this site, since I know many of the puzzlers here are brilliant. Based on the other problems in the book, I think concatenation (e.g. $ab$) represents multiplication as usual, even with spaces in between. I promise it's not homework, I just really want to know the solution! I've scanned the page and uploaded it; hopefully that doesn't breach copyright. Can anyone solve it?



A quick clarification, as there seems to have been some confusion expressed in the comments over the provenance of this puzzle. Basic of Madthematics is not a real book; the image is not a real scanned page (the text and background noise are different resolutions); the puzzle is in fact original and I therefore obviously know its solution.


I'd love to see this problem solved, so I'll put down my thoughts on it. So far, the bull has been noted; this seems to point to one way of interpreting the puzzle to devise a solution, i.e. interpreting the RHS components visually/semantically rather than mathematically. I suspect a form of cryptic clue plays a role here (and I think the wordplay/definition delimiter - and in some cases the definition itself - is so obvious as to make many of them trivial). One of the other 4 methods should be obvious from where I found the puzzle (in a maths textbook), and I think the other two will become clearer as you approach the solution, which I suspect is somewhat meta. I imagine it will be easier to pin down the exact solution once we have partial solutions by some of the 4 methods. Feel free to post partial answers!

Hint 2:

Since there has been very little progress on this puzzle so far, let me condense and reiterate the hints I have given. There are four methods by which the solution may be arrived at. Not all of these four are equally simple or obvious; in fact two of them will probably only be seen in retrospect when the solution is found by one of the other more obvious two. One of these is to interpret the puzzle purely mathematically. To do this, simply evaluate the given formulae in the most obvious mathematical way (which has not actually been done yet, although @PhysicsNoob has evaluated some of the lines). The resulting expression will have to be rearranged into a form which will give you the answer.

The second method is to interpret the RHS of the puzzle visually/semantically. The method in this case is not the same for each line. Some lines (like the bull) have an obvious meaning by this method. Others involve a simple cryptic clue-like interpretation of the RHS; the 'wordplay'/definition delimiter is in these cases included in plain sight, and in some cases the definition is so literal as to be trivial (note the solutions involved are not necessarily whole words, but could be letters).

Once you have partial solutions by these two methods, you will need to find a way to rearrange the mathematical expression, such that they become the same four word answer (bearing in mind that there will have to also be two other ways in which this answer can be obtained from the puzzle presentation; one of these two is similar to one of the others). The method at this step has a high dependence on wordplay (sometimes highly fragmented and phonetically approximate). That is the method of solution of this puzzle, which can theoretically be solved in four ways. Can you devise a solution?

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    $\begingroup$ Is there a possibility that the formatting is important? e.g. the section on X, E, O and I seem to look like a cow (or horse) head.... $\endgroup$ – APrough May 15 '20 at 13:11
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    $\begingroup$ Well the four lines starting with X, E, O and arrow make a drawing of a cow with horns. That's as far as I got... maybe it's just "a load of bull" ;) $\endgroup$ – Dmitry Kamenetsky May 16 '20 at 7:13
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    $\begingroup$ Maybe it's just me, but I found the flavour text a bit off-putting. It implies this is a puzzle taken out of context from the middle of a book, which might put people off if they feel there's potentially background information missing, or other puzzles in the book which build up to this one which might suggest a starting place. Making it clearer that this is an original puzzle might help it get more traction? $\endgroup$ – Mohirl Jun 15 '20 at 11:31
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    $\begingroup$ To add some balance to the comments here, personally I thought it was fairly clear all along that this was an original puzzle just playing on the trope of people coming to SE for advice (for one thing the book title is entirely fictional and turns up no Google search results) - especially having experienced some of the OP's other recent puzzles first-hand I've now got a sense of your style! Not to say that I think anything less of people who were fooled by it (I don't), but I just felt somebody ought to say that your effort to add interesting flavour-in-disguise was appreciated here! :) $\endgroup$ – Stiv Jun 15 '20 at 13:03
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    $\begingroup$ As a counter to melfnt, I'm now more interested in looking at it! I didn't look too deeply into the puzzle at first and thought the story was real (only having skimmed it). $\endgroup$ – Deusovi Jun 15 '20 at 17:42

Before starting off, I just want to state that I don't know the proper formatting, so the answer might be downvoted. Also, I might be pulling at straws for the most part, because I am just an engineering student trying to apply his brain at some puzzles, but here goes nothing-

My first guess is that all (or at least some) alphabets have a number associated with them. Keeping that in mind, a determinant for T can be calculated.

That comes out to be 2. Coincidently, a determinant is represented as Δ, and so that might prove useful when calculating the value of A. The thing in brackets above the matrix might just be the power (when turned to numbers) that we need to raise this Δ to.

Next, in R, I surmise that

[oz gal] might be a reference to "ounces (oz) in a gallon (gal)". That will come out to be 128. The reason why I don't think it is "gallons in ounces" instead is because that would be decimals, and who likes those! :P But then, keeping 1/128 in the back of the head doesn't hurt since S might be large, balancing out the 1/128.

I have some thoughts regarding 1ST as well (sorry, don't know how to write that).

It is a kind of a curveball...in a sense. Well, it should denote two things. a.) What is reads, first, and b.) 1^(SxT). In general, the superscript on 1 is in small letters. The use of capital letters, however, makes me think it has something to do with raising the power of 1, which would be 1 of course. As I said, kind of a curveball.

Someone might be curious as to how is my first interpretation of 1ST useful. Well,

Look at the P row. Doesn't that read like "1st rotate (and then the number inside tells the index)"?

Some of the findings like this are below-

T = 2^(3x1xAx↕)

R = 1x128xS OR R = S/128

E = 0

O = \x/

I conjecture that S is a square number. You see if the square root of S still belongs to the set of Natural Numbers (second last line), it has to be a square number. Similarly in the last line, S^(5/2) kinda points in the same direction. Also, the bar above might indicate taking an average of all the five elements of that set.

If _ is equal to 1, that would mean X is either equal to 1, or 1111, as X is just a bunch of underscores written in succession, which can be interpreted as 1's multiplying, or representing a number as it stands.

Lastly, the partial derivatives of Z added up, it reminds me of the gradient operation. Also, these partial derivative terms are natural numbers in themselves, as they are part of the set according to the second last line as well.

That's it for now, I will update if I think of anything else.

  • $\begingroup$ I'm really glad to see someone finally have a go at this puzzle! I really think with a systematic approach it will be quite amenable to devising a solution. My most important advice would be this: I've told you there are $4$ ways to find the solution. I've told you the two easiest ways are $(1)$ to solve the problem as a face-value mathematics problem and $(2)$ to approach the lines from a rebus/wordplay/cryptic clue point of view. In this answer, you're mostly following the former mathematical approach. Based on what you've written, I think you have the maths knowledge to complete this. $\endgroup$ – Anon Jul 14 '20 at 11:40
  • $\begingroup$ Now you just need to be systematic: Every line tells the value of a variable, except the second-last (which tells you - albeit in nonstandard notation - some facts about variables) and the first, which represents a formula to be evaluated. Try approaching this as an algebra problem and evaluate this first line and you will have made good progress towards a solution. Note that in this mathematical approach, there is nothing tricksy... Your interpretation oz gal = 128 would not work because mathematically oz gal does not mean 'number of ounces in a gallon'; oz gal = 128 oz oz would work. $\endgroup$ – Anon Jul 14 '20 at 11:40

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