Mind your decisions, I'm Presh Talwalkar
A mathematician writes numbers on a cube. On each face, the mathematician writes a positive integer.
For clarification, a positive integer is a whole number greater than zero.
On each vertex, the mathematician writes the product of its three adjacent faces.
For example in the upper right corner
The mathematician would write the number that's equal to the product of x,y,z.
The mathematician does this for each of the vertices.
If the sum of all vertices is 154, what is the sum of all faces?
A similar problem was asked to eighth-grade students in Taiwan, who are age roughly 13 to 14 years old.
Thanks to Kieran in New Zealand for the suggestion
Pause the video if you'd like to give this problem a try. And when you're ready keep watching to learn how to solve this problem
So, how can we solve this problem
suppose the faces are labeled a,b,c,d,e and f
Suppose the face we're looking at is labeled a we'll rotate the cube to label the faces b, c and d
We'll rotate the cube again to get back to a
on the left side, we'll label it as e and we'll label the face on the right as f
Let's now label each vertex. This vertex will be labeled abe
Then if we rotate the cube, this vertex will be labeled abf
Let's continue doing this for all the vertices
now on the bottom, we have the face d and on the left side, we have e
We still have d on the bottom and on the right side we have f. so we can label these two vertices
They will be ade and adf
Let's put these labels to the side
And now let's label the remaining four vertices which will be on the opposite side of the cube
so we'll look at face c and we'll label each of these vertices
On the left side will be e and on the right side will be face f
On the top of this will be face d. And on the bottom of c will be face b
We can use these labels to label each of these vertices
Starting from the lower left and going clockwise. We have be, cde, cdf and cbf
We want the sum of all of the vertices
Let's sum the first four and the second four separately
The first four we can factor out an a and the second four we can factor out a c
Notice the remaining bracketed terms are exactly equal in both of these equations
So we can further factor the sum as a + c multiplied by the bracketed term
Let's continue factoring
The first two terms here have a common factor of b
The next two terms have a common factor of d
We then have a common factor of e + f
So we get the following factorization for the sum of all these vertices
It'll be a + c multiplied by b + d multiplied by e + f
now recall this is equal to 154
since each number is a positive integer each bracketed term must be 2 or larger
And must be a factor of 154
154 factors into 2 times 7 times 11
Therefore, the three bracketed terms
must be equal to 2, 7 and 11, and the sum of the bracketed terms will be the sum of 2, 7 and 11
Therefore we have a + c plus b + d plus e + f is equal to 2 + 7 + 11
Now if we look at these bracketed terms, we have one of each letter
Therefore we have the sum of the faces and this is equal to 20 and that's the answer
The sum of the numbers on the faces is equal to 20. What a neat little problem
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