The “Battle of English and Mathematics” word problem has been making a name for itself around the internet recently. Some people claim the answer is 5, others 31. In actuality, this problem has 28 solutions, but we think the “correct” answer is 17.
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What is The Battle of English and Mathematics question?
Although word problems do sometimes get mistranslated as they pass around, the most frequently quoted “version” of this riddle seems to be the original. This is fortunate as the wording is important and very specific:
1 Rabbit saw 6 elephants while going to the river.
Every elephant saw 2 monkeys going towards the river.
Every monkey holds 1 parrot in their hands.
Q: How many animals are going to the river?
It should be noted that a difference between “to” and “toward” does exist but is irrelevant to the answer, so don’t think about it too hard.
Before moving on to the next section, try the puzzle yourself and remember the answer you get; it will come in handy later.
Why are people saying 5 is the answer?
The reason for getting 5 is likely due to how we think about math problems. Most people went to school at some point in their life, and were generally given quizzes and tests with one correct answer. This is a standard in the field of education, but it leaves some unintended side effects. Namely, growing up in such an environment fosters an expectation that “whenever I’m given a question, I’m expected to find the answer. Once I find an answer that works, I’m done.” This is not always true in the natural world, as some questions will have multiple answers.
The reasoning behind 5, typically goes like this: “1 rabbit saw 6 elephants while going to the river.” Only the rabbit was said to be going to the river, so the elephants must not be. “Every elephant saw 2 monkeys going to the river.” It didn’t say every elephant saw 2 different monkeys, therefore they must have seen the same two. “Each monkey holds 1 parrot in their hands. There are 2 monkeys, therefore 2 parrots. 1 rabbit + 2 monkeys + 2 parrots = 5 animals total.“
It seems logical, right? But remember, that’s based on the expectation that the question contains “the” answer. If this were a problem in English alone, that would be true. Nevertheless, the title is ‘The Battle of English and Mathematics’. “1 + 2 + 2 = 5” is not the mathematics portion of the question, as basic arithmetic is used steadily across every subject and field of study—just as how the inclusion of words alone does not make something an English problem.
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How to read the question from both perspectives: English (wording) & Math (logic)
Whenever asked a question or riddle (unless it is a trick question), you can expect that all the information needed to solve it is given. This guarantees the solution won’t be something like “there are more animals than were mentioned,” or “one of the parrots flew away.” However, this does not mean that uncertainties shouldn’t be accounted for. To explain this further, I’ve broken the riddle into pieces:
“1 Rabbit saw 6 elephants while going to the river. “ – We know based on the wording that there is exactly 1 rabbit and that it is going to the river. We know that there are exactly 6 elephants. We do not know whether or not the elephants are going to the river.
“Every elephant saw 2 monkeys going towards the river.” – We know that each elephant saw exactly 2 monkeys and that all monkeys are going to the river. We do not know if the elephants all saw the same monkeys or separate monkeys.
“Every monkey holds 1 parrot in their hands.” – We know that each monkey is holding exactly 1 parrot. We do not know if the monkeys are holding separate parrots, or the same parrot.
“How many animals are going to the river?” – We know what types of animals there are. We’re trying to solve for the total number of those animals that are going to the river.
This problem is worded to be purposely deceiving at first glance. To find the answer, we need to use language skills—the English portion of the problem. However, we are also purposely not given the total number of animals. To find the total, we need to use probability—the mathematics section of the question.
We will need to use both English and mathematics to find the solution.
Finding the real answer to the English and Mathematics problem:
If you’ve been reading so far, you might be wondering how such a question can be solved. Luckily, there is a “correct” answer! (And no, it’s not “anything” or “there isn’t enough information.” It’s a real answer—a nice answer—but it will require a few short steps to find.)
You may recognize this type of grid. Perhaps you remember it being referred to as a ‘Punnett Square” from biology, as a way to guess certain genotypes? If not, don’t worry. We won’t be looking at any gametes.
As it turns out, this type of grid is very useful for easily measuring probabilities. Allow me to explain how it works and why on Earth I’m using it to solve an internet meme.
These grids take all the possible values for one variable (how many of a type of animal could be going to the river), and add them to all the possibilities of another variable (another animal.) The result is a lovely grid that lists every possible outcome, without us having to find them all one-by-one. This is not the most effective way to do this, but certainly simple, and it gets the job done.
Because we have different possible numbers of parrots, monkeys, or elephants (explained earlier.) We’ll want to figure out the most likely number of total animals. To do this, we’ll take two groups of animals, parrots and monkeys, and plug them into the grid.
How many monkeys can there be? Well, if every elephant sees the same two monkeys, that would be 2 monkeys, so that’s our low number. Now, our highest number is if every elephant sees two completely different monkeys, and that would be 12 monkeys. Therefore, we have between 2 and 12 monkeys (all the various combinations make up for the numbers in between.)
What about parrots? Our lowest is if every monkey holds onto the same parrot, 1 (which would be weird, but possible.) Our highest is if we have 12 monkeys, and every monkey holds a different parrot, 12. (There’s another rule for parrots, but we’ll mention them later to cut back on stress.)
I’ve gone ahead and plotted the monkeys and parrots. Numbers above the grid are possible parrots, numbers beside the grid are possible monkeys. You can think of the grid itself as a battleship game, just instead of letters and numbers, we use two sets of numbers.
If you’re wondering where the black numbers come from, they’re the total number of animals for each combination. For example, the outcome of 4 monkeys and 2 parrots is 4 + 2 = 6. Now, because there is always 1 rabbit, we can add 1 to every single square (outcome.) 4 + 2 + 1 is 7. Therefore, if you go to “4 blue, 2 green” you’ll find a 7.
There you have it! With just that one simple square, we’ve calculated every single possible combination of parrots, monkeys, and rabbits! Of course, we’re not done yet; we’re forgetting about the elephants (ironic.)
How do we add elephants to the graph we already made? The answer is actually quite simple. Each of our animals (except rabbits) so far take up 1 dimension each, giving the grid 2 dimensions (x and y if you’re familiar with coordinates.) Therefore, all we have to do is add a 3rd dimension (z) for the elephants.
You may have noticed that the added dimension turned our square into a cube. Because ‘none of’ or ‘all of’ the elephants could be going to the river, that gives us 0 to 6—our new row. You can think of each elephant as adding a “layer” to the cube. Every time you go back a layer (up 1 elephant) the total number of animals (the combination of parrots, monkeys, and rabbits) goes up by 1 elephant.
The way I’ve laid out the square is so that the smallest number starts at the front-left-bottom (4), and steadily goes up until reaching the back-right-top (31).
Before we tally up our final result, we have to do one more thing. If you’ll recall, “Every monkey holds 1 parrot in their hands.” Although many monkeys can hold the same parrot, one monkey cannot hold multiple parrots. This means the total number of parrots can never exceed that of the monkeys.
To reflect this rule in our graph, we simply have to remove all instances where the # of parrots is greater than the # of monkeys. The end result is a wedge shape that now accurately portrays every combination. At this point, all that’s left is to count the “blocks.”
The final answer to The Battle of English and Mathematics problem:
Unless otherwise stated (such as by the “rules” of a standardized testing system,) uncertainties should always be considered as valuable information.
By taking this into account, the mathematical portion of the problem becomes one of simple probability—not basic arithmetic. To find the answer, we’ve compiled a list of outcomes from 4 animals, to 31.
The higher the bar for a certain number of animals, the more outcomes there are that result in that number. More outcomes = higher probability.
By taking a look at the graph above, there is one answer in particular that clearly has a higher probability than any other (winning 8.2% of the time, much higher than the average of 3.6%.) According to these results, the most likely number of animals going towards the river is 17.
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If I were teaching at a university and gave this question on an exam (I wouldn’t—I’m not that cruel), 17 is the answer I would be hoping to see. Then again, that’s only because I’d expect my students to examine the question in the same manner I did. After all, this is my interpretation as a science educator, whereas a mathematician or linguist may expect something completely different. As with all things in life, we can only apply what we know. My experiences tell me to consider all uncertainties in my answer, but depending on who asks, the “best” answer to give might be 5.
Without inferences, most riddles are unsolvable. How you make these inferences will rely heavily on your situation:If given as a question on an English test, it would be wise to infer the answer is 5.If asked on a job questionnaire for data analysis, it may be best to say 17.If given as a philosophical thought experiment, you might say “there can not be a decisive answer”…
One answer is not enough to deny the others, yet there is always an urge to “crack the code.” In this manner, riddles such as The Battle of English and Mathematics do a great job of exposing the vast gap between how we currently interpret the world around us, and what we have the potential yet to understand.
Hopefully, you gained something from this article
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