The goat problem (or bull-tethering problem) considers a fenced circular field of radius with a goat (or bull, or other animal) tied to a point on the interior or exterior of the fence by means of a tether of length , and asks for the solution to various problems concerning how much of the field can be grazed.Tieing a goat to a point on the interior of the fence with radius 1 using a chain of length , consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection(1)Taking gives(2)plotted above. Setting (i.e., half of ) leads to the equation(3)which cannot be solved exactly, but which has approximate solution(4)(OEIS A133731).Now instead consider tieing the goat to the exterior of the fence (or equivalently, to the exterior of a silo whose horizontal cross section is a circle) with radius . Assume that , so that the goat is not..
Rubik's Cube is a cube in which the 26 subcubes on the outside are internally hinged in such a way that rotation (by a quarter turn in either direction or a half turn) is possible in any plane of cubes. Each of the six sides is painted a distinct color, and the goal of the puzzle is to return the cube to a state in which each side has a single color after it has been randomized by repeated rotations. The puzzle was invented in the 1970s by the Hungarian Ernő Rubik and sold millions of copies worldwide over the next decade.The number of possible positions of Rubik's Cube is(Turner and Gold 1985, Schönert). Hoey showed using the Cauchy-Frobenius Lemma that there are positions up to conjugacy by whole-cube symmetries.The group of operations on Rubik's Cube is known as Rubik's group, and the Cayley graph of that group is called Rubik's graph. The minimum number of turns required to solve the cube from an arbitrary starting position is equal to the graph..
Rubik's Clock is a puzzle consisting of 18 small clocks, 14 of which are independent, each of which may be set to any 12-hour position. There are therefore possible configurations in total. The God's number (i.e., the graph diameter of the graph corresponding to Rubik's Clock, which is the minimum number of moves required to solve it from an arbitrary starting position-i.e., in the worst case) was shown by Kogler to be 12 (Kogler 2014; cube20.org).The numbers of positions from which the clock can be solved in , 1, ... moves are 1, 330, 51651, 4947912, 317141342, 14054473232, 428862722294, 8621633953202, 101600180118726, 528107928328516, 613251601892918, and 31893880879492, 39248 (A256586; cube20.org), which sum to as they must.
Given a straight segment of track of length , add a small segment so that the track bows into a circular arc. Find the maximum displacement of the bowed track. The Pythagorean theorem gives(1)But is simply , so(2)Solving (1) and (2) for gives(3)Expressing the length of the arc in terms of the centralangle,(4)(5)(6)(7)But is given by(8)so plugging in gives(9)(10)This is a transcendental equation that cannot be solved exactly with a closed-form solution for , but for ,(11)Therefore,(12)(13)Keeping only terms to order ,(14)(15)so(16)and(17)If we take and 1 foot, then feet. Solving equation (◇) numerically, we find that the true answer is feet.
A sultan has granted a commoner a chance to marry one of his daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter's dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must immediately decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will allow the marriage to take place only if the commoner picks the daughter with the overall highest dowry. Then what is the commoner's best strategy, assuming he knows nothing about the distribution of dowries (Mosteller 1987)?Since the commoner knows nothing about the distribution of the dowries, the best strategy is to wait until a certain number of daughters have been presented, then pick the highest dowry thereafter (Havil 2003, p. 134). The exact number to skip is determined by the condition that the..
Sudoku (literally, "single number"), sometimes also is a pencil-and-paper logic puzzle whose goal is to complete a grid satisfying various constraints. In the "classic" Sudoku, a square is divided into "regions", with various squares filled with "givens." Valid solutions use each of the numbers 1-9 exactly once within each row, column and region. This kind of sudoku is therefore a particular case of a Latin square.Under the U.S.-only trademarked name "Number Place," Sudoku was first published anonymously by Garns (1979) for Dell Pencil Puzzles. In 1984, the puzzle was used by Nikoli with the Japan-only trademarked name Sudoku (Su = number, Doku = single). Due to the trademark issues, in Japan, the puzzle became well-known as nanpure, or Number Place, often using the English name. Outside Japan, the Japanese name predominates.The puzzle received a large amount of attention in the..
How far can a stack of books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible for books (in terms of book lengths) is half the th partial sum of the harmonic series.This is given explicitly by(1)where is a harmonic number. The first few values are(2)(3)(4)(5)(OEIS A001008 and A002805).When considering the stacking of a deck of 52 cards so that maximum overhang occurs, the total amount of overhang achieved after sliding over 51 cards leaving the bottom one fixed is(6)(7)(8)(Derbyshire 2004, p. 6).In order to find the number of stacked books required to obtain book-lengths of overhang, solve the equation for , and take the ceiling function. For , 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (OEIS A014537) books are needed.When more than one book or card can be used per level, the problem becomes much more complex. For..
A well-known nursery rhyme states, "As I was going to St. Ives, I met a man with seven wives. Every wife had seven sacks, every sack had seven cats, every cat had seven kitts. Kitts, cats, sacks, wives, how many were going to St. Ives?" Upon being presented with this conundrum, most readers begin furiously adding and multiplying numbers in order to calculate the total quantity of objects mentioned. However, the problem is a trick question. Since the man and his wives, sacks, etc. were met by the narrator on the way to St. Ives, they were in fact leaving--not going to--St. Ives. The number going to St. Ives is therefore "at least one" (the narrator), but might be more since the problem doesn't mention if the narrator is alone.Should a diligent reader nevertheless wish to calculate the sum total of kitts, cats, sacks, wives, plus the man himself, the answer is easily given by the geometric series(1)with..
The "15 puzzle" is a sliding square puzzle commonly (but incorrectly) attributed to Sam Loyd. However, research by Slocum and Sonneveld (2006) has revealed that Sam Loyd did not invent the 15 puzzle and had nothing to do with promoting or popularizing it. The puzzle craze that was created by the 15 puzzle began in January 1880 in the United States and in April in Europe and ended by July 1880. Loyd first claimed in 1891 that he invented the puzzle, and he continued until his death a 20 year campaign to falsely take credit for the puzzle. The actual inventor was Noyes Chapman, the Postmaster of Canastota, New York, and he applied for a patent in March 1880.The 15 puzzle consists of 15 squares numbered from 1 to 15 that are placed in a box leaving one position out of the 16 empty. The goal is to reposition the squares from a given arbitrary starting arrangement by sliding them one at a time into the configuration shown above. For some initial arrangements,..
A cipher is an algorithm that converts data (plaintext) to an obfuscated form that is not directly readable. Ciphers are usually used with the intention of hiding the contents of a message or document from unauthorized persons. Ciphers can also be used to verify identity on the Internet.Cipher algorithms usually require a special "key" that can be used to encrypt the message. Usually, the key provides sufficient information for easy decryption of the ciphertext, however, some ciphers require a different keys for decryption. These algorithms are termed "asymmetric" key ciphers.Most ciphers are reversible, but there do exist algorithms that are non-reversible, termed one-way ciphers, or trap-door algorithms. These are commonly used for comparing passwords: the correct password is encrypted with the algorithm. Passwords that are then entered are encrypted with the same algorithm. The encrypted forms are then..
Caesar's method is an encryption scheme involving shifting an alphabet (so , , , , etc., ,,). It is one of the most basic encryption methods, and is a specialized form of a transposition cipher.Because the alphabet is rotated, the shift is consistent. A mapping of ,, etc. is termed ROT 1 because the letters are shifted by one. Note that would map to due to the circular aspect of the rotation.Encrypting with ROT and ROT yields the same ciphertext if and only if (mod 26).ROT 13 is a popular encryption method on the Internet since it obfuscates text sufficiently to prevent accidental reading of the text. It is commonly used to list answers of puzzles, for example.Caesar's method is a very weak encryption scheme, since there are 26 possibilities (of which one, ROT 26, is trivial). These can be easily checked by hand...