superprismatic

Here are the first 1001 digits of pi: 3. 14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679 82148086513282306647093844609550582231725359408128 48111745028410270193852110555964462294895493038196 44288109756659334461284756482337867831652712019091 45648566923460348610454326648213393607260249141273 72458700660631558817488152092096282925409171536436 78925903600113305305488204665213841469519415116094 33057270365759591953092186117381932611793105118548 07446237996274956735188575272489122793818301194912 98336733624406566430860213949463952247371907021798 60943702770539217176293176752384674818467669405132 00056812714526356082778577134275778960917363717872 14684409012249534301465495853710507922796892589235 42019956112129021960864034418159813629774771309960 51870721134999999837297804995105973173281609631859 50244594553469083026425223082533446850352619311881 71010003137838752886587533208381420617177669147303 59825349042875546873115956286388235378759375195778 18577805321712268066130019278766111959092164201989 We are to convert this into a string of letters. We will use the assignment A=0, B=1, C=2, ..., Z=25 and these rules for making strings: Let d(i) be the ith digit of pi (we ignore the decimal point). We start with an empty string and an index, n, initialized to 1. 1. if d(n) is in the set {0,3,4,5,6,7,8,9}, we concatenate the letter corresponding to d(n) to the string, then we increment n by 1. 2. if d(n) is 1 then we have the choice: 2a. concatinate B to our string, then increment n by 1. 2b. concatenate the letter corresponding to 10+d(n+1), then increment n by 2. 3. if d(n) is 2 and d(n+1) is greater than 5, we concatinate C to the string, then increment n by 1. 4. if d(n) is 2 and d(n+1) is less than 6, we have the choice: 4a. concatinate C to the string, then increment n by 1. 4b. concatinate the letter corresponding to 20+d(n+1) to the string, then increment n by 2. The challenge is to determine how many different strings we can make in this way from the first 1001 digits of pi. [/code] [b] Note well: It may seem that this problem is a good one for a programmer. Not so! The number of strings are so numerous that you cannot enumerate them with a program. The real solution to this problem rests with insight. Owing to the size of the data set (1001 digits), it may be helpful to massage the data using a program, but this is by no means necessary, and it may actually increase the time to solution. [/b]

Good try, but you can get a bit lower! By the way, use spoilers so others can look at the posts without seeing answers which might spoil their experiences working on the problems. Another thing: You can use the code /code tags to make sure your columns align correctly.

Take the integers from 1 to 13 and distribute them among 4 sets such that the sum of the products of the contents of the sets is minimal. Each set must be non-empty. If a set contains only one number, that number is considered to be the product for that set. For example, if I divided the numbers as follows: {1,2,3,4,5,7}, {6,11,12}, {8,9,10}, and {13}, their products would be 840, 792, 720, and 13 respectively. The sum of these is 2365. So, 2365 is the sum of products for this particular way of distributing the 13 numbers into 4 sets. Two questions: 1. What is the minimum value for the sum of products? 2. In how many different ways can you achieve the minimum?

The problem begins to be easily seen as you begin with the third color:

Redrooster, please check your original post for typos in the data. If you find any, I'd like to re-run my program. After all, the way I did things, one bad data point would spell disaster!

Here's my work so far: I decided to write a program to see if any code digit depends on a particular subset of the box identifier positions (the letter with the four digits). So, I looped over all subsets of Box identifiers. For example, while processing the subset consisting of the first, third, and fourth box identifier positions, I would only consider the E39 from the Box identifier E1390. Then, I would look thru all other Box identifiers which had an E, 3, and 9 in any order. Then I would count which positions in the code for the 2 boxes did not change. If I found that a particular code position changed when the subset of the first, third, and fourth positions in two Box identifiers were the same, I would conclude that that particular code position did not depend solely depend on the first, third, and fourth positions. I did this for all nonempty subsets of Box identifier positions, all pairs of the 77 box identifier & code instances, and all code positions. I found that there were 11 instances where positions 3, 4, and 5 of one Box identifier had the same subset of numbers as that of another Box identifier. In all 11 cases, their first code numbers matched. We already knew that the first code position depended on the numbers in positions 3, 4, and 5, however, it was a good test of my program that it found that. The bad news is that I found no other code position for which this kind of agreement happened once, let alone 11 times, with any subset of Box identifier positions. It seems to me, then, that the individual code positions (except the first) are not functions of any subset of Box identifier positions. So, I think we should start looking at ways to produce the last three digits as a single entity from the Box identifiers. I hope someone duplicates my work, as this would validate (or possibly invalidate) my claim. I will gladly offer any help or answer any questions about this problem. Redrooster: be assured that some of us are working on this problem, but it's difficult to report on negative results.

Well, just call me 'Saber Skunk' from now on 'cause I misread the clue in post #9 to say 'positive' instead of 'non-negative' which really cut me to the quick!

I think your count is too small.

I just wrote a quickie program to get

I found a simple-to-generate cyclic 7920 pattern that contains this sequence but there are no words in it as you describe. In my generation scheme, the next 3 terms (after the 5) are 6, 26, 17. Perhaps you could supply more terms.

Yes, I made a pretty big mistake when I tried to get around the problem of the absolute value function being non-differentiable. Thanks for pointing out my error. I didn't read the explanation in your first post carefully enough.

Great problem! Hard, but fun to work. Thanks, KS!

There seems to be disagreement about the answer. So,

I take it that the numbers have no leading zeros. Am I correct?

Will you tell us how you arrived at the solution? Please do so.

Solve the following set of 10 nonlinear equations in 10 positive integer variables A,B,C,D,E,F,G,H,I,J: B(CE+A)+AI = 33 AB(CGH+1)+AI = 99 (FH+1)AI+BCE = 1413 H(AFI+J+ABCG) = 1464 J(DFI+H)+AB = 320 (A+E)I+ACDF+HJ = 180 H(AFI+J)+EI+D(H+ACF) = 1569 BC(AGH+E)+I(DFJ+E)+HJ = 413 AC(DF+BGH) = 204 H(ABCG+J)+BCE = 84 [/code] Here, as is usual, multiplication is assumed when terms are concatenated. Note also that the left hand sides of both the 2[sup]nd[/sup] and 3[sup]rd[/sup] equations contain a 1 (I don't want anyone to confuse these 1s with Is). This system has a unique solution in the positive integers. If need be, I will post hints every several days to get people started on the way to solve this one.

You are quite good at this, ljb!

Impressively well done, ljb!