On Thinking Outside the Box (Column 656)
With God’s help
A few weeks ago I saw, in one of the tabloids distributed in synagogues, the following riddle in an ad for the Lev Academic Center.
At the bottom they wrote that the solution indicates that the solver “thinks outside the box.” This prompted some gloomy reflections about what “thinking outside the box” actually means. Is solving a riddle like this really outside-the-box thinking?
Analysis
First, let’s try to solve the riddle. My assumption is that each shape represents some number. There are three shapes here—sphere, pyramid, and cube—each stands for a number. An additional assumption is that each spatial position/relationship between shapes represents a mathematical operation between the numbers they stand for. We have two positions: a shape above a shape, and a shape (or stack of shapes) next to a shape (stack). Each such relation represents some mathematical operation (which may or may not be the same).
At first glance I assume the numbers should be integers, and the operations should be taken from the basic/simple mathematical operations (addition, subtraction, multiplication, division, and perhaps exponentiation). Why? Because that seems reasonable and natural.
With these assumptions, the solution comes rather naturally. Start with the top equation. It has two identical “stacks,” each made of two shapes. It therefore makes sense to begin by decoding the “next to” relation. Should we multiply the two stacks, add them together, or subtract them? Simple elimination shows it can’t be multiplication, since 24 has no integer square root. It also can’t be subtraction, because then the result would have to be 0 and not 24. Nor can it be exponentiation, since there’s no integer that solves XX=24X^X=24. What remains, then, is addition.
This immediately yields two preliminary results: (1) placing stacks next to each other denotes addition; and (2) the value of each stack in the top equation is 12. Now we must ask what “a shape above a shape” means. As noted, it could be any of the five operations listed above (including addition itself). A first proposal is that it’s a different operation, since different spatial configurations encode different mathematical operations. Perhaps it is exponentiation—but only if we assume the cube is 12 and the pyramid is 1. On that assumption, the bottom equation gives the sphere as −5, and then we get 12+(−5+1)=712 + (−5 + 1) = 7. According to this, in the third equation we obtain (−5)12=1,955,078,125(−5)^{12} = 1,955,078,125.
Well, that can be ruled out, because the bottom of the riddle shows that the result must give us a day of the month, i.e., an integer between 1 and 31. This also rules out the infinite set of solutions with non-integers. We must therefore return to our assumptions.
Note that the instructions for solving the riddle speak of three equations (exercises?), where the third must be solved using the first two. Seemingly the riddle is self-contained, and there is no reference to the hint at the bottom of the picture; it’s merely the result of the solution and not a hint. Yet if the result must indeed be an integer day, that does rule out the solution I gave above—so I must use that as a hint as well. It seems the riddle’s wording is imprecise.
All right, so it’s natural to try decoding “shape above shape” as multiplication. In that case, if the numbers are integers, it’s reasonable—checking the top equation—to set the cube to 4 and the pyramid to 3 (or vice versa). Returning to the second equation, in both cases the circle is 1. But then the result of the third equation changes: it will be either 4 or 3 respectively. Both are whole numbers that yield a calendar date. Those two dates this year fall on Wednesday and Thursday, so neither can be ruled out (I assume Lev won’t hold an open day on Shabbat). We’re left with two different options, both of which fit. There’s no way to decide between them. I presume this is a mistake by Lev—perhaps they didn’t think far enough outside the box (or maybe I missed something?).
By the way, this isn’t the only solution. Without the “day of the month” hint at the bottom, other solutions become possible. For example, still taking “above” to be multiplication, assign to the upper stacks the numbers 2 and 6. If the cube is 2 and the pyramid is 6, the circle below comes out non-integral (1.2), and the final result is 2.4—again not an integer. If we flip them—cube 6 and pyramid 2—then the circle is 0.5. That’s not integral either, but the final result becomes 6/0.5=126/0.5 = 12 (which this year falls on a Friday—legitimate for an open day).
We could also decode “shape above shape” as subtraction (and not multiplication). That opens many possibilities. Each of the upper stacks must yield 12. A subtraction that gives 12 could be, say, 18−618 – 6, i.e., cube 6 and pyramid 18. Using that in the second equation, the circle is 17, and the last equation yields −11. The hint rules this out.
A third option is to decode “shape above shape” as addition as well, just like “next to.” (Exactly as in a pair of equations with two variables XX and YY, there’s nothing that prevents us from getting a solution where X=YX=Y.) In that case there are 12 ways to place the cube and pyramid. If, for example, we take the cube to be 8 and the pyramid 4, the second equation gives the circle as −4, and the third equation yields 4—a perfectly legitimate result. I didn’t check further, but I assume there are additional solutions in this direction.
What if we decode the vertical stack as division? For example, cube 36 and pyramid 3, but then the final result isn’t an integer (we get 9). It seems the other outcomes here are also fractional, so this isn’t a viable option.
That should suffice before we consider what “outside-the-box” thinking might be.
Which solution is “outside the box”?
Even before I noticed the hint at the bottom, my first impression was that the solution was 3 or 4. My assumption was integers, with adjacency meaning addition, and the vertical relation being some other operation. I immediately assumed multiplication. As noted, this produces two different dates, yet it still seemed clear this was the intended answer, and I took the duplicity as their mistake. Suppose I got a reasonable, natural result—why assume it’s the correct one? Why not consider the possibility of identical operations (addition and addition), or addition and exponentiation? Even taking the hint into account, we still saw several other possibilities—for instance, allowing fractional values for the shapes so long as the final result is an integer between 1 and 31; alternatively, treating the vertical relation as addition and allowing negative values for the shapes so that the date comes out sensible. These two solutions may feel less natural and less “obvious,” but why conclude from that that they’re wrong?
I’d go further. The first solution I presented is the result of natural, expected reasoning—inside-the-box thinking. The other two actually give us outside-the-box solutions. In particular, those two yield a single date, whereas the “natural” solution yields two possible dates. So why am I still sure the intended solution is one of the “natural” ones? Simple: because it’s clear to me they expected an inside-the-box solution, not an outside one. Solutions with fractional or negative numbers are “outside the box,” and therefore not the ones they were waiting for.
The conclusion is that the Lev Academic Center is actually testing who thinks inside the box, not who thinks outside it—and perhaps we can even say they themselves think inside the box. They absolutely want systematic, orderly, good thinking; that is, they want the admitted student to be someone capable—but only so long as he thinks inside the box, not outside it. A person who thinks outside the box will arrive on 12.7 and won’t be admitted. The one who picks 4 via the second solution might show up on the right day yet still not get accepted (because in exams they filter out “outside-the-box” thinking), or show up on the wrong day and not get accepted. A person who thinks inside the box will conclude the result is either 3 or 4, and will therefore come on both days and get in. Their error will, at worst, make him work harder, but in the end—no worries—if he thinks sufficiently inside the box, he’ll be admitted.
To clarify better what “inside” and “outside” the box mean, let me turn to a familiar argument by Wittgenstein.
Wittgenstein on Rule-Following
I discussed this before (see, e.g., here, and also this PDF) about the famous “rule-following” argument. Suppose that on a psychometric exam you’re given the sequence:
…2, 4, 6, 8, 10 — and asked for the next term. The obvious answer is 12—i.e., the even numbers (with the recursive rule “add 2,” and then add 2 again, and so on). But Wittgenstein argues that the “next number” could, just as well, be anything—say 16, because the rule could be: add 2, add 2, add 2, then multiply by 2, then add 2 again, etc. Now consider the sequence …3, 5, 7. The natural response is 9, on the assumption we’re dealing with odds (or primes, in which case the next would be 11). But equally well it could be any other number you wish.
He goes further: any given numerical sequence can be continued in any way whatsoever; for every such continuation there exists some suitable rule. For example, take the second sequence 3, 5, 7… Suppose we find a student who wrote −4.7 as the next term. Has he necessarily made a mistake? Not at all. Here’s a rule that justifies his continuation.
Let the rule be a 3rd-degree polynomial with four coefficients:
f(n) = a1 + a2·n + a3·n^2 + a4·n^3
If we want the sequence to be 3, 5, 7, … we choose the coefficients accordingly, by setting up four equations in four unknowns:
f(1) = a1 + a2 + a3 + a4 = 3
f(2) = a1 + 2a2 + 4a3 + 8a4 = 5
f(3) = a1 + 3a2 + 9a3 + 27a4 = 7
f(4) = a1 + 4a2 + 16a3 + 64a4 = −4.7
Solving these is straightforward. The result is:
a1=14.7a_1 = 14.7, a2=−23.11666a_2 = -23.11666, a3=13.7a_3 = 13.7, a4=−2.28333a_4 = -2.28333.
Plugging back in gives:
f(n) = 14.7 − 23.11666·n + 13.7·n^2 − 2.28333·n^3
Substituting n=1,2,3,4n=1,2,3,4 yields exactly the desired values. In other words, the answer written by the student—−4.7—is just as “correct” as any other. This is, of course, only an example; one can do the same for any sequence and any desired continuation in countless ways (choose higher-degree polynomials, other functional forms, etc., tuning the parameters to the target).
Back to outside-the-box thinking
So why won’t the student who wrote −4.7 be admitted to university? Because he thinks outside the box. The psychometric exam expects an inside-the-box answer, and sorts students accordingly—admitting those who think inside the box and filtering out those who think outside it.
You probably suspect I’m criticizing the academic filtering process, but I’m not. This is how it should work. Imagine the process allowed everyone who thinks like that to be admitted: the fellow who wrote −4.7, together with one who wrote 1, another who wrote 1549, a third who wrote 17.3, a fourth who wrote 2e, and so on. If every answer is acceptable, what exactly is the exam testing? How is any filtering being done? In such an exam, only a student who justifies his answer (provides an appropriate f(n)f(n)) would be admitted. Random answers without justification are out (this is not a multiple-choice test but a psychometric exam requiring full solutions).
Now the lecturer walks into class to teach some mathematics. He has no chance. You simply cannot teach such a class. Wittgenstein explains there that every rule we want to teach our students relies on giving examples and generalizing. Suppose we teach them to count the natural numbers in base 10—first from 1 to 10, then to 100, then 1000. The teacher asks a student to continue the count, and he answers without hesitation: e−. The astonished teacher asks how he got that, and the student produces an f(n)f(n) that yields the first thousand naturals followed by e−. Fine, the teacher presses on and teaches up to 10,000. The next student answers 15, and of course provides a function that “justifies” it. In fact, these students don’t need to supply the justifying functions—their brains are wired such that after 1,000, e− just feels natural to them, and to the other one after 10,000, 15 just feels right. This will happen to that teacher in any mathematical subject he tries to teach, elementary or advanced.
The conclusion is that when students’ minds are built in a peculiar way (outside the box), there’s no way to teach them. A teacher can teach only students whose minds are built like his own—i.e., those who have the same box he does. No wonder an institution constructed in a certain way wants to admit only students whose minds fit the institution’s standard. Students with a different mental makeup simply won’t be able to learn there. No wonder Einstein and other geniuses were known as “bad students”: their minds were built differently from their teachers’ and everyone else’s, so teaching them was very hard.
The upshot is that the Lev Academic Center is right in its policy of admitting those who think inside the box. The only question that bothers me is why they claim they’re looking specifically for those who think outside the box. Well, perhaps you need outside-the-box thinking to understand that.
You can find a charming story about “outside-the-box” thinking here.
References mentioned: mikyab.net post, and this TAU PDF: https://law.tau.ac.il/sites/law.tau.ac.il/files/media_server/law_heb/Law_Society_Culture/books/procedurot/ProceduresRiesenfeld.pdf.
Article Contents
With God’s help: On Thinking Outside the Box
A few weeks ago I saw, in one of the tabloids distributed in synagogues, the following riddle in an advertisement for Machon Lev:
Below they wrote that the solution shows that the solver thinks outside the box. This prompted some gloomy thoughts in me about the definition of thinking outside the box. Is solving such a riddle really thinking outside the box?
Analysis. First, let us try to solve the riddle. My assumption is that each shape represents some number. There are three shapes here—a ball, a pyramid, and a cube—each of which represents a number. A further assumption is that every position, or spatial relation, between shapes represents a mathematical operation between the numbers represented by them. We have here two positions: a shape above a shape, and a shape (or a structure of shapes) beside a shape (or structure). Each of these relations represents a mathematical operation, which may or may not be the same one. At first glance I assume that the numbers should be integers, and the operations should be among the basic and simple mathematical operations: addition, subtraction, multiplication, division, and perhaps exponentiation. Why? Because that seems reasonable and natural to me. Under these assumptions, when one approaches the riddle the solution is fairly obvious. Let us begin with the top equation. There are two identical structures there, each composed of two shapes. So it is reasonable to begin by deciphering the relation of standing side by side. Should the two structures be multiplied, added, or subtracted? Simple elimination shows us that this cannot be multiplication.
24 has no integer square root. Nor can it be subtraction, because then the result would have to be 0 and cannot be 24. Nor can it be exponentiation, because there is no integer that solves the equation. What remains, then, is addition. This conclusion immediately gives us two initial results: placing structures next to one another means addition, and the value of each of the two structures in the top equation is 12. We now have to ask ourselves what the meaning of a structure of one shape above another is. As noted, this could be any of the five operations I listed above, including addition itself. A first suggestion is that this is a different operation, since different spatial arrangements represent different mathematical operations. It could perhaps be exponentiation, but only if we assume that the cube is 12
and the pyramid is 1. On that basis, from the lower equation it follows that the ball is 5.
According to this, in the third equation we obtain 1,955,078,125.
Fine, but this result can be ruled out, since at the bottom of the riddle there appeared another line:
We see below that the result is supposed to give us a date in the month, that is, an integer between 1 and 31. This also excludes all the infinitely many possible solutions with non-integer numbers. We must therefore return to our assumptions. It is worth noting that the instructions for solving the riddle speak of three equations, or perhaps exercises, and say that the third is to be solved by means of the first two. Apparently the riddle stands on its own, and there is no reference to the hint at the bottom of the image; that line is only a result of the solution and not a hint. But if the result really must be an integer, that rules out the solution I suggested above. In other words, I must use this as a hint too. It seems to me that the riddle is formulated imprecisely here. Well then, if so, it is natural and obvious to try to interpret the relation of one shape above another as multiplication. In that case, if the numbers are integers, it is reasonable to check in the top equation that the cube is 4
and the pyramid is 3, or vice versa. If we return to the second equation, in both cases the circle is 1. But the result of the third equation changes: it will be 4
or 3, respectively, in the two cases. This is an integer that gives the date in the month. This year those two dates fall on Wednesday and Thursday, so neither can be ruled out. I assume Machon Lev will not hold an open day on the Sabbath. We are left with two different possibilities, both of which fit. There is of course no way to decide between them, and I assume this is Machon Lev’s mistake: they did not think far enough outside the box—or perhaps I missed something here. By the way, this is not the only solution. Were it not for the hint at the bottom of the picture, other possible solutions could also be suggested. For example, one can assume that this is indeed multiplication and place in each of the top structures the numbers 2 and 6. If the cube is 2
and the pyramid is 6.
then below we get that the circle is 1.2, which is not an integer. The final result comes out 2.4, again not an integer. And if we assume the opposite, that the cube is 6
and the pyramid is 2, then the circle is 0.5. This is admittedly not an integer, but the final result still comes out 6/0.5
(12), a number that could fit; this year it falls on Friday, which is legitimate for an open day at Machon Lev. One could also interpret the configuration of one shape above another as subtraction rather than multiplication. This of course gives us many possibilities. Each of the top structures has to give 12. A subtraction that gives us 12
could be, for example, 18–6.
That is, the cube is 6
and the pyramid is 18. If we use this in the second equation, the circle comes out to 17, and then the result of the final equation is 11. But the hint shows that this does not fit. A third possibility is to interpret the configuration of one shape above another as addition as well, just like the configuration of one shape next to another, exactly as in a pair of equations with two unknowns X and Y, where there is no obstacle to getting the same value for both.
In such a case there are several ways to assign the cube and the pyramid. If, for example, we take the cube as … and the pyramid as …, I get from the second equation that the circle is 4. And from the third equation we obtain the result 4, which is entirely legitimate. I did not check further, but I assume there are more solutions in this direction. What if we interpret the structure of two shapes, one above the other, as division? For example, the pyramid is 36
and the cube is 3. In such a case the circle is 9 1/3, but then the final result is not an integer. It seems to me that all the other results here also come out fractional, and so this is not an option. Good, enough of what we have seen so far. Let us move on and ask ourselves something about thinking outside the box.
What is the solution that is outside the box? Even before I noticed the hint at the bottom of the picture, at first glance it was clear to me that the solution was 3
or 4. My assumption was that these were integers, that standing side by side is addition, and that standing one above another is a different operation. I immediately assumed it was multiplication. As stated, this gave me two different results, but despite that it was clear to me that this was the answer they were looking for, and I assumed the duplication was their mistake. Suppose I obtained a sensible and natural result. Why assume that this is really the correct one? Why not take into account the possibility that these are identical operations—addition and addition—or addition and exponentiation? Even after taking the hint into account, we have still seen several other possibilities. For example, under the assumption that the values of the shapes may be fractional, so long as they lead us to a result that is an integer between 1 and 31. Alternatively, the relation of one shape above another could also be addition, while allowing negative values for the shapes, in a way that yields a sensible date. These two solutions may indeed seem to us less natural and less compelling, but why think that for that reason they are not correct?
I would say even more than that. The first solution I presented is the result of obvious, natural, and expected thinking—in other words, thinking inside the box. Precisely the two other results give us solutions that are outside the box. In particular, those two latter possibilities each give us a unique solution, whereas the natural solution gives us two different possible solutions. So why am I nevertheless sure that they intended one of the natural solutions? Simply because it is obvious to me that they were expecting precisely a solution inside the box and not outside it. Solutions with fractional or negative numbers are outside the box, and so those are not what they expected.
The conclusion is that the people at Machon Lev are trying here to test precisely who thinks inside the box and not who thinks outside it, and one might perhaps even say that they themselves think inside the box and not outside it. They certainly want systematic, orderly, and good thinking from you. In other words, they want the student they admit to be a person of ability, but only so long as he thinks inside the box and not outside it. A person who thinks outside the box will arrive on 12.7
and will not be admitted. The other person, who reaches some other date in the second solution, may perhaps come on the right day and still not be admitted, because in the tests they will filter him out for thinking outside the box; or he may come on the wrong day and not be admitted. A person who thinks inside the box will conclude that the result is either 3
or 4, and then he must come on both days and thus be admitted. Their mistake will at most cause him to work hard, but in the end, have no fear: if he thinks sufficiently inside the box, he will be admitted. To clarify more fully the meaning of thinking inside and outside the box, I will now turn to a well-known argument of Wittgenstein.
Wittgenstein on following rules. I have discussed here in the past, see for example column 482, and also here in Chapter 2, Wittgenstein’s well-known argument called ‘following a rule’. Suppose that in a psychometric exam you are presented with the following sequence: 2,4,6,8… and are asked what the next term in the sequence is. The obvious answer is 10
of course, because we assume that the rule is the sequence of even numbers, and the recurrence formula governing it is adding 2
from each term to the next. But Wittgenstein argues that the answer could just as well be 16, since the rule could be as follows: multiply by 2, add 2, add 2, and then multiply by 2
and so on. You understand that the answer could be any other number you like. Now think about the sequence 3,5,7… What is the next number? One naturally wants to write 9, on the assumption that this is the sequence of odd numbers. But it could just as well be 11, if the sequence is the primes, those divisible only by themselves and 1.
He goes on to make a more general claim. Any given sequence of numbers can be continued in any way you wish. Every such continuation has a justification by means of a suitable rule. For example, suppose we are given the second sequence, 3,5,7… and when we check one of the exams we discover a student who wrote -4.7 as the next term. Did he necessarily make a mistake? Not at all. Here is a rule that will justify this continuation. Suppose the rule is of the form of a polynomial with four coefficients: f(n) = a1 + a2n + a3n2 + a4n3. If we want the sequence f(n) to give us the desired sequence, we must determine the coefficients accordingly. To do so, we construct four equations with four unknowns: f(n=1) = a1 + a2 + a3 + a4 = 3; f(n=2) = a1 + 2a2 + 4a3 + 8a4 = 5; f(n=3) = a1 + 3a2 + 9a3 + 27a4 = 7
f(n=4) = a1 + 4a2 + 16a3 + 64a4 = -4.7. One can easily solve these four equations for the four coefficients. The result obtained is: a1 = 14.7; a2 = -23.11666; a3 = 13.7; a4 = -2.28333. If we substitute the result into the expression for f(n), we get the desired sequence:
f(n) = 14.7 – 23.11666n + 13.7n2 – 2.28333n3. If you now substitute n=1,2,3,4 into f, you get exactly the desired results. In other words, we have proved that the answer of that student, who wrote -4.7 as the continuation, is just as correct as any other answer. This is, of course, only an example, and it can be done for any sequence with any result you want, in countless different ways. One may choose a higher-degree polynomial sequence, or any other functional form, while fitting the parameters to the desired sequence.
Back to thinking outside the box. So why, nevertheless, will the student who wrote -4.7 not be admitted to study at the university? Because he thinks outside the box. The psychometric exam expects an answer that reflects thinking inside the box, and it sorts the students so that those who think inside the box are admitted and those who think outside it are filtered out. You presumably think I am writing this critically of the academic screening process, but not at all. That is indeed how it ought to operate. To understand this, think of a situation in which the process allows all the fellows who think in such ways to be admitted to study. That same chap who wrote -4.7, together with one who wrote 1, another who wrote some other number, a third who wrote 17.3, a fourth who wrote pi, and a fifth 2, and so on. You will ask: if every answer is accepted, what exactly does this exam test? How, if at all, is any screening carried out here? In this exam, only a student who provides a justification for his answer is admitted to study, namely an f(n) function
appropriate to the answer. Mere answers without justification are not accepted. This is not a multiple-choice exam, but a psychometric one that requires full answers. Now the lecturer enters the classroom and wants to teach them some mathematical topic. He has no chance of doing so. It is simply impossible to teach such a class. Wittgenstein explains there that every rule we want to teach our students is based on giving examples and generalization. For example, we want to teach them to count the natural numbers in the decimal system. We teach them to count from
to 10, and then to 100
and to 1000. Then the teacher instructs one of the students to continue the counting, and he answers without hesitation: -e. The astonished teacher asks him how he arrived at that answer. The student immediately supplies him with a function f(n), which gives us the first thousand natural numbers and after them -e. Fine. The teacher does not despair and continues teaching up to 10,000, but then the next student who is asked to continue answers 15, and of course supplies a suitable function to justify that. In fact, these students do not really produce the justifying functions. Their minds are built in such a way that after 1,000
for one of them, -e comes out naturally, and for the second, after 10,000
the number 15. This will happen to that teacher in every mathematical subject he tries to teach them, advanced or basic. The conclusion is that when the students’ minds are built in a strange way, outside the box, there is no way to teach them. A teacher can teach only students whose minds are built like his own, that is, those who have a box identical to his. It is no wonder that when an institution built in a certain way sorts students, it wants to admit only students whose minds are built according to the institution’s standard. Students with a different mental structure simply will not be able to study there. It is no wonder that Einstein and other geniuses were known as poor students. Their minds were built differently from the standard of their teachers and of everyone else, and therefore it is very hard to teach them. The conclusion is that Machon Lev is right in its policy of admitting precisely students who think inside the box. The only question that troubles me is why it lies and writes that it is specifically looking for those who think outside the box. Well, apparently one needs thinking outside the box in order to understand that.
Here you can find a nice story about thinking outside the box.
Discussion
In this comment I will represent the idiots among your readers, at least on the mathematical side: for people like us, when there are two operands standing one above the other vertically – it is obvious that the operator is division. The thought that it could be otherwise is itself thinking outside the box.
Einstein actually was a good student, at least in mathematics.
In general, everyone thinks inside his own box.
This is the definition of a “tabloid”:
“Yellow journalism (singular: tabloid) is a derogatory term for journalism perceived as low-quality, dealing mainly with gossip, sex, and scandals, including sex scandals, and appealing to the target audience’s emotions and instincts through sensationalist methods of writing, editing, and design.”
You may dislike the pamphlets in the synagogue, and they do have advertisements (standard ones) and a simple design (modest and unremarkable). Can you explain why to call them דווקא by such a term?
An interesting article.
As for the ad itself, I think that “thinking outside the box” here means not trying to set up the equations with the symbols as ordinary unknowns, but rather to combine them. An ordinary person will simply see two equations with three unknowns and perhaps not know how to proceed. The special kind of thinking is the ability to use tools that are not only technical mathematical ones, but also elimination and logic.
The explanation is simple: “thinking outside the box” is a marketing slogan. No attempt was made to tailor the slogan to the specific puzzle that appeared beside it. Nor is the intention that the institution is looking for people who think outside the box; the point is that this sentence makes a good impression on most of the public and will help attract students.
Einstein got satisfactory grades in his degree; he was definitely not among the outstanding students.
I got that the answer is 8: the number of faces of one level multiplied by the number of vertices of the second level, assuming the shapes are a cube, a cone, and a sphere, and assuming that a sphere has one face and a cone has 2.