1. Solve this cryptic equation, realizing of course that values for M and E could be interchanged. No leading zeros are allowed.
This can be solved through systematic application of logic. For example, cannot be equal to 0, since . That would make , but , which is not possible.Here is a slow brute-force method of solution that takes a few minutes on a relatively fast machine:
This gives the two solutions
777589 – 188106 == 589483
777589 – 188103 == 589486
Here is another solution using Mathematica’s Reduce command:
A faster (but slightly more obscure) piece of code is the following:
Faster still using the same approach (and requiring ~300 MB of memory):
Even faster using the same approach (that does not exclude leading zeros in the solution, but that can easily be weeded out at the end):
Here is an independent solution method that uses branch-and-prune techniques:
And the winner for overall fastest
2. Which of the following expresses Google’s over-arching philosophy?
A) “I’m feeling lucky”
B) “Don’t be evil”
C) “Oh, I already fixed that”
D) “You should never be more than 50 feet from food”
E) All of the above
[This exercise is left to the reader.]
3. This space left intentionally blank. Please fill it with something that improves upon emptiness.
For nearly 10,000 images of mathematical functions, see The Wolfram Functions Site visualization gallery .
4. What is the coolest hack you’ve ever written?
While there is no “correct” answer, a nice hack for solving the first problem in the SIAM hundred-dollar, hundred-digit challenge can be achieved by converting the limit into the strongly divergent series:and then using Mathematica’s numerical function SequenceLimit to trivially get the correct answer (to six digits),
You must tweak parameters a bit or write your own sequence limit to get all 10 digits.
5. What number comes next in the sequence: 10, 9, 60, 90, 70, 66, ?
A) 96
B) 1000000000000000000000000000000000
0000000000000000000000000000000000
000000000000000000000000000000000
C) Either of the above
D) None of the above
This can be looked up and found to be sequence A052196 in the On-Line Encyclopedia of Integer Sequences, which gives the largest positive integer whose English name has n letters. For example, the first few terms are ten, nine, sixty, ninety, seventy, sixty-six, ninety-six, ?. A more correct sequence might be ten, nine, sixty, googol, seventy, sixty-six, ninety-six, googolplex. And also note, incidentally, that the correct spelling of the mathematical term ”
googol” differs from the name of the company that made up this aptitude test.
6. Given a triangle ABC, how would you use only a compass and straight edge to find a point P such that triangles ABP, ACP and BCP have equal perimeters?
(Assume that ABC is constructed so that a solution does exist.)
This is the isoperimetric point , which is at the center of the larger Soddy circle. It is related to Apollonius’ problem . The three tangent circles are easy to construct: The circle around has diameter , which gives the other two circles. A summary of compass and straightedge constructions for the outer Soddy circle can be found in ” Apollonius’ Problem: A Study of Solutions and Their Connections” by David Gisch and Jason M. Ribando
7. ‘Tis known in refined company, that choosing K things out of N can be done in ways as many as choosing N minus K from N: I pick K, you the remaining.
This simply states the binomial coefficient identity .
Find though a cooler bijection, where you show a knack uncanny, of making your choices contain all K of mine. Oh, for pedantry: let K be no more than half N.’Tis more problematic to disentangle semantic meaning precise from the this paragraph of verbiage peculiar.
8.
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
What’s the next line?
312211. This is the “look and say” sequence in which each term after the first describes the previous term: one 1 (11); two 1s (21); one 2 and one 1 (1211); one 1, one 2, and two 1′s (111221); and so on. See the look and say sequence entry on MathWorld for a complete write-up and the algebraic form of a fascinating related quantity known as Conway’s constant.
9. What’s the next number in this sequence: 10, 9, 60, 90, 70, 66 … ?
A. Spell the numbers out:
Ten
Nine
Sixty
Ninety
Seventy
Sixty-six
They are in ascending order, based on the number of letters in the spelled-out numbers. A correct response will have nine letters: 96, for instance. A cleverer answer is “one googol.” That’s the huge number that can be written as a “1″ with a hundred zeros after it. Google, the company’s name, was
originally a misspelling of “googol.”
10. What will be the next great improvement in search technology?
Semantic searching of mathematical formulas. See http://functions.wolfram.com/About/ourvision.html for work currently underway at Wolfram Research that will be made available in the near future.
11. What’s broken with Unix?
Their reproductive capabilities.
How would you fix it?
[This exercise is left to the reader.]