你能做几道？Jane Street烧脑面试题！

如果你想在Jane Street（一家全球知名与神秘的做市商公司）找到一份收入最高的工作，你需要有解决一些极难的脑筋急转弯。Jane Street因在面试时向其候选人提出具有挑战性的脑筋急转弯而闻名。公众号列出几个给各位Quant瞧瞧，看看你们能做出来否，祝你好运！

Puzzle1

Each of the grids below is incomplete. Place numbers in some of the empty cells so that in total each grid’s interior contains one 1, two 2’s, etc., up to seven 7’s. Furthermore, each row and column within each grid must contain exactly 4 numbers which sum to 20. Finally, the numbered cells must form a connected region, but every 2-by-2 subsquare in the completed grid must contain at least one empty cell.

Some numbers have been placed inside each grid. Additionally, some blue numbers have been placed outside of the grids. These blue numbers indicate the first value seen in the corresponding row or column when looking into the grid from that location.

Once each of the grids is complete, create a 7-by-7 grid by “adding” the four grids’ interiors together (as if they were 7-by-7 matrices). The answer to this month’s puzzle is the sum of the squares of the values in this final grid.

最新一期，答案未公布

Puzzle2

A random line segment of length D is chosen on a plane marked with an infinite checkerboard grid (i.e., a unit side length square grid).  What length D maximizes the probability that the segment crosses exactly one line on the checkerboard grid, and what is this maximal probability?

答案：

The answer to this puzzle is that D = 1 and the resulting probability is 2/pi (or about .6366). 

Puzzle3

Call a “ring” of circles a collection of six circles of equal radius, say r, whose centers lie on the six vertices of a regular hexagon with side length 2r. This makes each circle tangent to its two neighbors, and we can call the center of the regular hexagon the “center” of the ring of circles. If we are given a circle C, what is the maximum proportion of the area of that circle we can cover with rings of circles entirely contained within C that all are mutually disjoint and share the same center?

When submitting an answer, you can either send in a closed-form solution, or your answer out to 6 decimal places.

答案

The answer when rounded to 6 decimal places is 0.783464.

Puzzle4

Two friends, Alter and Nate, have a conversation:

Alter: Nate, let’s play a game. I’ll pick an integer between 1 and 10 (inclusive), then you’ll pick an integer between 1 and 10 (inclusive), and then I’ll go again, then you’ll go again, and so on and so forth. We’ll keep adding our numbers together to make a running total. And whoever makes the running total be greater than or equal to 100 loses. You go first.

Nate: That’s not fair! Whenever I pick a number X, you’ll just pick 11-X, and then I’ll always get stuck with 99 and I’ll make the total go greater than 100.

Alter: Ok fine. New rule then, no one can pick a number that would make the sum of that number and the previous number equal to 11. You still go first. Now can we play?

Nate: Um… sure.

Who wins, and what is their strategy?

答案

Nate (the first player) can always win in this game, by starting with the number 3.After this first turn, Nate can force the running total to increment by units of 12.This could happen 2 different ways:

1、If Alter picks some number X between 2 and 10, Nate chooses 12-X

2、If Alter picks 1, Nate responds by picking 1 as well.  Now Alter cannot pick 10 (since this would force the sum of the previous two numbers to be 11), and must pick some other number Y.  Nate then picks 10-Y.

In this way, Nate can force Alter to choose numbers when the running total is equal to 3, 15, 27, 39, 51, 63, 75, 87, and 99.  At this point, Alter is forced to take the total to 100 or greater.

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