You're right, Allan, I have problems in defining that rule -- which seems obvious when you work with it, but hard to explain (please someone, help!-) Here are a few examples, taken from my search of the "ten blocks" maximum weight: I started with 1,2,3,4,5,6,7,8,9,10 -- untouched, sum 55 then 10,9,8,7,6,5,4,3,2,1 -- blocks must be reorganized: try: 10, 98, 765, 4321 -- sum 5194 (I cannot reorganize in 109, 876, 5432, 1 as the "pieces" are not increasing from left to right -- BUT I cannot either reorganize in 10, 987654321 with sum 987654331 because Alice will tell me: "Too easy, pal! There is a way to break that in "minimal pieces!" And she would be right, of course with the solution above: 10, 98, 765, 4321 -- sum 5194. In short, Alice will always try to show me that the block ordering I've selected (among a huge lot of others -- factorial 10) can be "disassembled" to reduce the sum I was so proud of (the first constraint being still ok) Dunno if this helps... Best, É.
Le 26 octobre 2019 à 16:08, Allan Wechsler <acwacw@gmail.com> a écrit :
Please clarify the "minimal pieces" constraint? Everything seems to depend on that, and I cannot infer what it is from the examples given.
On Sat, Oct 26, 2019, 6:16 AM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, Alice and Bob play the "Chunk & Sum" game. Which consist in A giving to B an integer and asking B to chunk it in smaller pieces that must be increasing from left to right. Example: 3223584222 is given by A to B. B chunks it like this: 3 22 358 4222. The "weight" of a partition is given by the sum of the pieces: 3 + 22 + 358 + 4222 = 4605 The aim of B is to produce the highest weight. The aim of A is to warn B, because his partition must obey a precise "minimal pieces" rule: "Hey, Bob, your partition is bad, this is the right one: 3 + 22 + 35 + 84 + 222 = 366". B agrees, his first partition forgot the "minimal pieces" way to chunk.
After a while, Alice and Bob have an idea: to give to each other wooden blocks instead of integers -- blocks with numbers printed on them. A receives the blocks marked 1, 2, 3, 4, 5. B receives the blocks marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The game consists now to put one's blocks in a certain order on the table such to produce a number to chunk – this number having the highest weight according to the two rules we have just seen: 1) pieces in increasing order from left to right 2) "minimal pieces". Alice forms the integer 43521 with her blocks and claims that among the 120 possible solutions, this is the best one. Indeed, the only way to properly chunk 43521 is 43 + 521 = 564 and 564 is the maximum you can reach. What is the integer Bob should form with his set of blocks in order to maximize the weight? ____________________ (warning: this is hard to compute by hand, my best result is over 100,000 -- took me the night) (note that the 1 and the 0 of the block marked 10 cannot be separated) This idea could produce a nice little sequence for the OEIS: "Highest possible weight that one can reach with blocks marked 1 to n, according to the Chunk & Sum game". I guess the sequence would start like this: 1, 21, 33, 325, 564,... (not in the OEIS, if the hereunder computations are ok) n = 1 --> integer produced is 1 with weight = 1 n = 2 --> integer produced is 21 with weight = 21 n = 3 --> integer produced is 231 with weight = 2 + 31 = 33 n = 4 --> integer produced is 4321 with weight = 4 + 321 = 325 n = 5 --> integer produced is 43521 with weight = 43 + 521 = 564 etc. My solution for n = 10 is here, with Daniel Day-Lewis, on my personal blog: https://bit.ly/31KVedK) Best, É.
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