superprismatic

Consider the following array of numbers: 8 0 -4 -7 5 5 -11 11 14 -11 3 9 -4 -13 14 1 2 1 7 -6 14 -3 -3 3 12 -10 -14 12 -11 10 -4 -4 -13 -11 8 -12 1 14 3 -12 -1 -3 2 2 4 -10 -8 -5 -9 2 11 11 -10 5 -10 -12 [/code] Your task is to pick elements out of this array so that the sum of the elements you pick is as large as possible. You may pick the same element multiple times as long as you don't violate any of the following rules. Also, in the following, "between" is always inclusive. OVERALL RULE: You must choose precisely 100 elements out of the array. ROW RULES: You must choose between 4 and 30 elements from the first row, between 7 and 39 from the second row, between 8 and 20 from the third, between 6 and 22 from the fourth, between 7 and 35 from the fifth, between 4 and 29 from the sixth, and between 2 and 47 from the seventh. COLUMN RULES: You must choose between 1 and 31 from the first column, between 3 and 42 from the second, between 7 and 36 from the third, between 7 and 40 from the fourth, between 5 and 30 from the fifth, between 1 and 34 from the sixth, between 1 and 9 from the seventh, and between 3 and 38 from the eight. INDIVIDUAL ELEMENT RULES: In the following array, the element [A,B] in the i[sup]th[/sup] row and j[sup]th[/sup] column specifies that you must choose the element located in the i[sup]th[/sup] row and j[sup]th[/sup] column of the original array between A and B times. [code] [0,11] [0, 9] [0, 4] [2, 9] [2, 4] [2,12] [1, 9] [1, 4] [1, 8] [0, 2] [0, 9] [1, 3] [1,13] [2, 9] [0, 8] [1, 7] [0,12] [1,13] [2,14] [1, 9] [0, 0] [2,14] [2, 8] [1, 1] [1, 8] [1, 7] [2, 3] [0,12] [1, 8] [0,14] [1, 2] [1,13] [0, 4] [1, 3] [2,12] [1, 3] [1, 8] [1,11] [0, 2] [2, 4] [1, 8] [0,11] [1, 2] [0, 5] [0, 6] [0, 1] [1,14] [1,10] [2, 3] [0,11] [0,12] [0,10] [1,13] [2, 6] [0, 3] [0, 7] So, for example, the -7 in the first row of the original array must be chosen at least twice but no more than 9 times. I have checked several times that it is possible to make choices which follow all the rules. I have a putative best answer. See how large your sum of 100 elements can be while following all the rules. I'll let you know if you get close (or surpass) my best answer.

Yep.

The 26 letters of the alphabet are written in scrambled order in a circle. A message is then enciphered using the following scheme. Each letter of the message is represented by two letters whose (clockwise) distance apart in the circle is the position of the letter in the alphabet. Any two consecutive letters, say PJ, could represent A, JP representing Y; any doubled letter could represent Z, et cetera. A message enciphered in this way becomes QJ XE BN JA ZB RK RJ EZ NA CA JE XQ EZ XR BQ CX KZ NA OB JS AO QN BX CX KJ KS AX OC XV BV ZU FV FJ UM XL GL [/code]What does the message say? (Begins "A complete")

I am having some troubles reducing this to a macro on excel though: that is the only knowledge in programming that I acquired (by reading the help file for visual basic!!!!!). So don't keep your hopes up Well, you have correctly counted the number of 4 subsets of 3 each from 12. But the problem only requires 4 non-empty subsets. So, they could be of sizes 7, 2, 2, and 1, for example. You are reading too much into the problem. You will be exhausting over only 15,400 of the 611,501 possibilities!

Yes, there are that many. The number is a Sterling number of the second kind and counts these things. Wikipedia has a good article on it at http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind

Nice! Do you mind giving a quick overview of how you did it?

Yeah, I thought of a guy humming a slowish tune and roughly seeing where the fish is on each beat. I actually originally made some bad assumptions which I had to abandon before I could solve it. I thought the fun of this problem was its intentional vagueness. This makes it a bit more fun because I am used to solving math problems where the hypotheses are very unambiguous. So, when I succeeded with this one, I felt an uncharacteristically weird sense of accomplishment. I hope you have as much fun with it as I did!

All correct. Now the fish could get to a diagonal square other ways than by crossing over the vertex. I assume that the person writing down the sequence would adjust the moments slightly until the fish is pretty clearly within a square. The fish may even be in the same square at two successive moments, either by not leaving it or by leaving and returning to it before the next "moment" expires.

The bottom of a pool is marked off in two rows of squares. The letters A to M in order are written in one row and the letters N to Z in order in the other. A goldfish swims aimlessly in the pool and the letter it is over at any moment is noted. A sequence of letters formed in this way is used as key to encipher a message (by mod-26 addition). The result is: FHTYI NYXKU QHDXS SHJIW BYOPN CLUR [/code]What was the message (contains the word "brave")? SUPERPRISMATIC NOTE: As usual with these puzzles, spaces in the cipher are only there for ease of reading. Note also that Penney did not say what the value of the letters are. So, the solver must make some assumption about that. It is part of the puzzle. Of course, knowing that it is solvable means that some wild scheme of assigning numbers to letters could not have been used. So, it may be safe to say that the numerical value of letters increases by 1 for each successive letter in the alphabet. But, is A=0? Or is A=1? Does it matter? That's part of the fun in solving this! Furthermore, successive "moments" must not be too far apart lest the key be so random as to make the problem unsolvable. You need to make some reasonable assumption as to how this restricts the key -- more solving fun!

They are not necessarily in order so that 1,3,4; 2,5,7;... would be valid.

I've tried to construct a puzzle which would be in the Walter Penney style but which would be a bit of a challenge to the programmers out there. I hope many of you rise to the challenge! The numbers from 1 to 12 are divided into 4 non-empty subsets. Each subset has its members put in order from smallest to largest. Each has a pointer initially pointing to the smallest member of the subset. The sum of the numbers pointed to are used to encrypt a letter of a message. Then, each pointer is moved forward (circularly) pointing to the next number in its subset. Encryption continues in this way until the entire message is encrypted. Encryption is accomplished by advancing the letter of the message as many places along the standard alphabetical sequence (also circularly) as the sum of the pointed-to numbers specify. Thus, if the sum of the pointed-to numbers were 33 and the message letter were Q, then the cipher letter would be X. Read this message (spaces are not important -- they just make the message easier to read): VKGTI YRKCP OQGCH OSFFA UIDYK UYNOV HQGYR QDFML MSHTA IFLAM VQGAE QY [/code]

Sorry, I misstated the problem. In order to guarantee being able to shoot perpendiculars, the triangle needs to be acute. So, the problem should read: Consider an acute triangle with vertices at (0,0), (a,0), and (b,c) with 0 < b < a. Consider a point (x,y) inside this triangle. Shoot perpendiculars from (x,y) to each side of the triangle. This divides the triangle into 3 quadrilaterals. Find (x,y) such that these quadrilaterals all have the same area.

Consider the triangle with vertices at (0,0), (a,0), and (b,c) with 0 < b < a. Consider a point (x,y) inside this triangle. Shoot perpendiculars from (x,y) to each side of the triangle. This divides the triangle into 3 quadrilaterals. Find (x,y) such that these quadrilaterals all have the same area.

Rs is just a unit for money used in INDIA ,so thats not important at all. The real thing that is puzzling me is that in all the 10 cases(mentioned in my prev post) the difference is coming out to be 1300 and i dont have a mathematical reason for this. So if some one can prove why this happening via some equation i will be grateful to him. It will also help me understand on which kind of equations similar phenomena of constant difference occurs. Thank You Anurag Kesarwani Consider the pair (P,A) where P is the stock price and A is the total monetary accumulation trader X has so far in his trading. This pair starts off to be (100,0) at the beginning of the first day. Let D be the function which takes (P,A) to its new value at the end of the day if the stock falls 10 points. We have that D[(P,A)]=(P-10,A+100-10P). Similarly, if we let U be the function which takes (P,A) to its new value at the end of the day if the stock rises 10 points. We have that U[(P,A)]=(P+10,A+100+10P). So, U[D[(P,A)]]=U[(P-10,A+100-10P)]=((P-10)+10,(A+100-10P)+100+10(P-10))=(P,A+100) [/code] and, [code] D[U[(P,A)]]=D[(P+10,A+100+10P)]=((P+10)-10,(A+100+10P)+100-10(p+10))=(P,A+100) which shows that U[D[(P,A)]]=D[U[(P,A)]] [/code] which tells us that D and U commute, as functions, with each other. So, in your example of 3 Ups and 2 Downs, the order doesn't matter. With a=110, you will always end with the same accumulation (Rs 1300). If you require more of an explanation, feel free to ask and I will try my best to explain further.

Six words were written one below the other in a 6x6 square. The numbers 0 through 5 were written in scrambled order down the side and the words shifted to the right (cyclically) these amounts. The same sequence of numbers was written across the top of the resulting square and the various columns shifted down (cyclically) these amounts. The result was: Y R O O N Y O L A O P O A S T N T C E S T U R M I E N E S G B S P E R R [/code] What were the words?

Yes, wrap around applies.

(1) The "scrambled sequence" in the first sentence is a scramble of the alphabet. (2) the spaces in the string starting with KDZAS are there only to make transcription easier. (3) each letter of the "scrambled sequence" is advanced (4) the next letter of the plain is encrypted.

Each letter of a message is replaced by its right-hand neighbor in a certain scrambled sequence. After each step all the letters in this scrambled sequence are advanced one position in the normal alphabetical sequence. (A follows Z.) With this scheme BALTIMORE becomes ZGQZYRBAM. What does KDZAS EBKOK GZYEV YJTWQ LBXJY WI represent, using the same initial set-up?