MathNovatory/Applied Maths/Game Maths/Chain


The Chain

Enigma
This game, as well as Finding The Faulty Coin and Weighing, comes from an enigma which furnishes valuable mathematical information. Here is an excellent enigma, furnishing all the required information with the least detail. As in Weighing, all the whole numbers from 1 to 23 must be produced and the number of opened links (2) is not given.

Specific To General
If "n" represents the number of links opened, the total number of links in the chain will be (((n + 1) x (2(n + 1))) - 1).
     If 1 link is opened (n = 1), there will be 7 links in the chain ((2 x 22) - 1).
     If 2 links are opened (n = 2), there will be 23 links in the chain ((3 x 23) - 1).
     If 3 links are opened (n = 3), there will be 63 links in the chain ((4 x 24) - 1).
To know which links to open, we must know the sizes of the sections produced.
     If 1 link is opened, the first section must have 2 links, we open the 3rd link,
          the 2nd section will have 4 links.
     If 2 links are opened, the first section must have 3 links, we open the 4th link,
          the 2nd section will have 6 links, we open the 11th link,
               the 3nd section will have 12 links.
     If 3 links are opened, the first section must have 4 links, we open the 5th link,
          the 2nd section will have 8 links, we open the 14th link,
               the 3nd section will have 16 links, we open the 31st link,
                    the 4th section will have 32 links.

Problem-solving
As in Weighing, this is also a case of logic in the sense that there is only one unique solution. Here again, reasoning the answer is not as simple as it was in the case of Finding The Faulty Coin (with its 9 = 32). We must here again build up the process from scratch (in the previous paragraph) and see which solution gives a total of 23 (by opening 2 links). We still consider this a planified, orderly, if slightly fastidious process rather than trial-and-error.