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| In [[computer science]] and [[mathematics]], the '''Josephus Problem''' (or '''Josephus permutation''') is a theoretical problem related to a certain [[counting-out game]].
| | A tenancy-in-widespread agreement is used whenever there may be widespread possession of a property. A person might go for actual property investment along with his siblings, buddies, or another get together. In any of those instances, it is a good option. It describes how the property is owned and what's the responsibility of each proprietor. That is the easiest way to divide the property with none problems.<br><br>b) A unit in an authorised condominium development. Take a look at our characteristic properties for a list of landed home for rent with photos and price. For a listing of serviced residences available, click on right here Investing within the property market is like playing the monopoly board recreation. It is advisable to have the proper strategies with the intention to achieve success. What higher technique to Resources">grasp the property investment strategies than by studying from the specialists and real property insiders. Factors that determine Singapore's property market A Look Back on 2011 and Wanting Forward to 2012 (at Propwise.sg) Each property has an AV. This AV of a property is estimated primarily based on market rentals of comparable or comparables properties. Tax Charge Previous to 1 Jan 2014 2.How is property tax calculated?<br><br>When you owner-occupy your own home, you'll be able to apply for owner-occupier tax charges If you're a Singaporean or Singapore Permanent Resident who personal and occupy just one private residential property or govt condominium, you need not apply because the concession will likely be granted routinely. It is easy to search out the best property agent to characterize you at HirePropertyAgent! Simply describe the kind of property, dimension, location and the expectations you need of the [http://www.sonomasoroptimist.org/?q=node/349290 singapore property agent] (eg. fee price, specialist in your space, experience, etc) Founder and chief executive of Second Likelihood Properties Mohamed Salleh Marican instructed The Straits Instances that each property class in Singapore appears to be coming off its peak price levels. Getting ready your property on the market<br><br>Its aroma acts as a stimulant. It is carminative, which implies it helps in relieving from gas. It is used to deal with many stomach-related problems. This natural drugs is used towards nausea, dyspepsia, flatulence, catarrh, rheumatism, and enteritis. It additionally has tonic and antibacterial properties. As a consequence of these two properties it's used in veterinary and homeopathic medicine. Its root can be utilized for treating fungal infections and in pain reduction of joints and muscles.<br><br>Goodrich stated ninety nine % of his shoppers buy for investment and so they search for yields of around 3 to 4 p.c. The highest 1 percent are on the lookout for a trophy property. Property management includes widespread prices and utilities. Guidelines concerning the precise usage of the property by the co-house owners together with issues like pets, noise levels, ground covering, and so forth., and responsibility of every co-proprietor for property upkeep. Other than the aforementioned factors, it additionally includes particulars concerning dispute decision, mortgage of the property, sale of property, proper to lease, enhancements and repairs, and termination and signatures. Within the Rest of Central Region , property costs were up by 0.9%. Outdoors Central Area , property costs rose by 3.8% q-q. 14xxpsf view this mission<br><br>But currently they have been transferring up market, brokers say. The current in-vogue building among the many Chinese language is Central Park's One57, a brand new skyscraper designed by Pritzker Prize-profitable French architect Christian de Portzamparc, the place they will spend $18.eighty five million for a 3-bedroom or $fifty five million for an house taking on your complete 81st floor. The constructing comes with all of the amenities of a 5-star resort. Shanghai businessman Wang Jiguang has already picked up two houses in one other major Australian metropolis, Melbourne, and one condo in Sydney. "My youngster is going to check abroad, and we are simply getting ready some abroad belongings for our baby, which will probably be much less risky," Wang said in a telephone interview from Shanghai. Uses in Hair and Skin Care<br><br>For properties owned by more than one owner, all house owners are collectively liable for paying property tax. For correspondence purpose, we are going to normally tackle to the owner who is listed first within the Notice of Transfer filed by the vendor's lawyer. Therefore, it's important that you as the new owner or one of the new homeowners, inform the seller's lawyer the one that will be responsible for the property tax issues. As soon as the switch record is up to date, we will likely be corresponding with the individual on all property tax issues, including payment of property tax. |
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| There are people standing in a [[circle]] waiting to be executed. The counting out begins at some point in the circle and proceeds around the circle in a fixed direction. In each step, a certain number of people are skipped and the next person is executed. The elimination proceeds around the circle (which is becoming smaller and smaller as the executed people are removed), until only the last person remains, who is given freedom.
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| The task is to choose the place in the initial circle so that you are the last one remaining and so survive.
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| ==History==
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| The problem is named after [[Flavius Josephus]], a Jewish historian living in the 1st century. According to Josephus' account of the [[siege of Yodfat]], he and his 40 soldiers were trapped in a cave, the exit of which was blocked by [[Roman soldiers|Romans]]. They chose suicide over capture and decided that they would form a circle and start killing themselves using a step of three. Josephus states that by luck or possibly by the hand of God, he and another man remained the last and gave up to the Romans.
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| The reference comes from Book 3, Chapter 8, par 7 of Josephus' ''[[The Jewish War]]'' (writing of himself in the third person):
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| {{quotation|However, in this extreme distress, he was not destitute of his usual sagacity; but trusting himself to the providence of God, he put his life into hazard [in the manner following]: "And now," said he, "since it is resolved among you that you will die, come on, let us commit our mutual deaths to determination by lot. He whom the lot falls to first, let him be killed by him that hath the second lot, and thus fortune shall make its progress through us all; nor shall any of us perish by his own right hand, for it would be unfair if, when the rest are gone, somebody should repent and save himself." This proposal appeared to them to be very just; and when he had prevailed with them to determine this matter by lots, he drew one of the lots for himself also. He who had the first lot laid his neck bare to him that had the next, as supposing that the general would die among them immediately; for they thought death, if Josephus might but die with them, was sweeter than life; yet was he with another left to the last, whether we must say it happened so by chance, or whether by the providence of God. And as he was very desirous neither to be condemned by the lot, nor, if he had been left to the last, to imbrue his right hand in the blood of his countrymen, he persuaded him to trust his fidelity to him, and to live as well as himself.<ref name="Josephus18">[[Flavius Josephus]], [http://www.gutenberg.org/dirs/2/8/5/2850/2850-h/book3.htm#2HCH0008 Wars Of The Jews III. 8. 7.] (Translation by William Whiston).</ref>}}
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| ==Solution==
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| In the following, <math>n</math> denotes the number of people in the initial circle, and <math>k</math> denotes the count for each step, that is, <math>k-1</math> people are skipped and the <math>k</math>-th is executed. The people in the circle are numbered from <math>1</math> to <math>n</math>.
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| ===k=2===
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| We explicitly solve the problem when every 2nd person will be killed, i.e. <math>k=2</math>. (For the more general case <math>k\neq 2</math>, we outline a solution below.)
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| We express the solution recursively. Let <math>f(n)</math> denote the position of the survivor when there are initially <math>n</math> people (and <math>k=2</math>).
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| The first time around the circle, all of the even-numbered people die.
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| The second time around the circle, the new 2nd person dies, then the new 4th person, etc.; it's as though there were no first time around the circle.
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| If the initial number of people was even, then the person in position <math>x</math> during the second time around the circle was originally in position <math>2x - 1</math> (for every choice of <math>x</math>). Let <math>n=2j</math>. The person at <math>f(j)</math> who will now survive was originally in position <math>2f(j) - 1</math>. This gives us the recurrence
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| :<math>f(2j)=2f(j)-1\;.</math>
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| If the initial number of people was odd, then we think of person 1 as dying at the end of the first time around the circle. Again, during the second time around the circle, the new 2nd person dies, then the new 4th person, etc.
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| In this case, the person in position <math>x</math> was originally in position <math>2x+1</math>. This gives us the recurrence
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| :<math>f(2j+1)=2f(j)+1\;.</math>
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| When we tabulate the values of <math>n</math> and <math>f(n)</math> we see a pattern:
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| {|
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| |<math>n</math> || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
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| |-
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| |<math>f(n)</math> || 1 || 1 || 3 || 1 || 3 || 5 || 7 || 1 || 3 || 5 || 7 || 9 || 11 || 13 || 15 || 1
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| |}
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| This suggests that <math>f(n)</math> is an increasing odd sequence that restarts with <math>f(n)=1</math> whenever the index ''n'' is a power of 2.
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| Therefore, if we choose ''m'' and ''l'' so that <math>n=2^m+l</math> and <math>0\leq l<2^m</math>, then <math>f(n)=2 \cdot l+1</math>.
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| It is clear that values in the table satisfy this equation. Or we can think that after <math>l</math> people are dead there are only <math>2^m</math> people and we go to the <math>2l+1</math>th person. He must be the survivor. So <math>f(n)=2l+1</math>. Below, we give a proof by induction.
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| '''Theorem:''' If <math>n=2^m+l</math> and <math>0\leq l<2^m</math>, then <math>f(n) = 2l+1</math>.
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| '''Proof:''' We use [[strong induction]] on <math>n</math>. The base case <math>n=1</math> is true.
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| We consider separately the cases when <math>n</math> is even and when <math>n</math> is odd.
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| If <math>n</math> is even, then choose <math>l_1</math> and <math>m_1</math> such that <math>n/2 = 2^{m_1}+l_1</math> and <math>0\leq l_1 < 2^{m_1}</math>. Note that <math>l_1 = l/2</math>.
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| We have <math>f(n) = 2f(n/2)-1=2((2l_1)+1) - 1=2l+1</math>, where the second equality follows from the induction hypothesis.
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| If <math>n</math> is odd, then choose <math>l_1</math> and <math>m_1</math> such that <math>(n-1)/2 = 2^{m_1}+l_1</math> and <math>0\leq l_1 < 2^{m_1}</math>. Note that <math>l_1 = (l-1)/2</math>.
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| We have <math>f(n) = 2f((n-1)/2)+1=2((2l_1)+1) + 1=2l+1</math>, where the second equality follows from the induction hypothesis. This completes the proof.
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| We can solve for <math>l</math> to get an explicit expression for <math>f(n)</math>:
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| :<math>f(n) = 2(n-2^{\lfloor log_2(n) \rfloor})+1</math>
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| The most elegant form of the answer involves the binary representation of size <math>n</math>: <math>f(n)</math> can be obtained by a one-bit left cyclic shift of <math>n</math> itself. If we represent <math>n</math> in binary as <math>n=1 b_1 b_2 b_3\dots b_m</math>, then the solution is given by <math>f(n)=b_1 b_2 b_3 \dots b_m 1</math>. The proof of this follows from the representation of <math>n</math> as <math>2^m+l</math> or from the above expression for <math>f(n)</math>. | |
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| '''Implementation:''' A simple python function can be:
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| <syntaxhighlight lang="python">
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| def josephus_2( n ):
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| from math import log
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| return 2*(n - 2**(int(log(n,2))))+1</syntaxhighlight>
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| ===The general case===
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| The easiest way to solve this problem in the general case is to use [[dynamic programming]] by performing the first step and then using the solution of the remaining problem. When the index starts from one, then the person at <math>s</math> shifts from the first person is in position <math>((s-1)\bmod n)+1</math>, where n is the total number of persons. Let <math>f(n,k)</math> denote the position of the survivor. After the <math>k</math>-th person is killed, we're left with a circle of <math>n-1</math>, and we start the next count with the person whose number in the original problem was <math>(k\bmod n)+1</math>. The position of the survivor in the remaining circle would be <math>f(n-1,k)</math> if we start counting at <math>1</math>; shifting this to account for the fact that we're starting at <math>(k\bmod n)+1</math> yields the recurrence
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| : <math>f(n,k)=((f(n-1,k)+k-1) \bmod n)+1,\text{ with }f(1,k)=1\,,</math>
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| which takes the simpler form | |
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| : <math>g(n,k)=(g(n-1,k)+k) \bmod n,\text{ with }g(1,k)=0</math>
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| if we number the positions from <math>0</math> to <math>n-1</math> instead.
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| This approach has [[Big-O notation|running time]] <math>O(n)</math>, but for small <math>k</math> and large <math>n</math> there is another approach. The second approach also uses dynamic programming but has running time <math>O(k\log n)</math>. It is based on considering killing ''k''-th, 2''k''-th, ..., <math>(\lfloor n/k \rfloor k)</math>-th people as one step, then changing the numbering.
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| '''Implementation:''' A simple python function can be (but might be slow/fail for large n and small k):
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| <syntaxhighlight lang="python">
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| def josephus(n, k):
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| if n ==1:
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| return 1
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| else:
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| return ((josephus(n-1,k)+k-1) % n)+1
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| </syntaxhighlight>
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| Alternatively a simple loop can also be run like:
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| <syntaxhighlight lang="python">
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| def josephus(n, k):
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| r = 0
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| i = 1
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| while i <= n:
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| r = (r + k) % i;
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| i+= 1
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| return r+1
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| </syntaxhighlight>
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| A dynamic programming solution in Haskell:
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| <syntaxhighlight lang="haskell">
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| josephus n k = josephus' [1..n] k where
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| josephus' xs k
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| | length xs == 1 = head xs
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| | otherwise = josephus' (killnext k xs) k where
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| killnext k xs = take (length xs - 1) (drop k (cycle xs))
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| </syntaxhighlight>
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| A somewhat more elaborate version in Java:
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| <syntaxhighlight lang="java">
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| int josephus(int n, int k) {
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| return josephus(n, k, 1);
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| }
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| int josephus(int n, int k, int startingPoint) {
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| if(n == 1)
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| return 1;
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| int newSp = (startingPoint + k - 2) % n + 1;
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|
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| int survivor = josephus(n - 1, k, newSp);
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| if (survivor < newSp) {
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| return survivor;
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| } else
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| return survivor + 1;
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| }
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| </syntaxhighlight>
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| ==Variants and generalizations==
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| According to ''[[Concrete Mathematics]]'', section 1.3, Josephus had an accomplice; the problem was then to find the places of the two last remaining survivors (whose conspiracy would ensure their survival).
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| ===Extended Josephus problem===
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| Problem definition: There are ''n'' persons, numbered 1 to ''n'', around a circle.
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| We eliminate second of every two remaining persons until one person remains. Given the ''n'', determine the number of ''x''th person who is eliminated.<ref>Armin Shams-Baragh [http://www.cs.man.ac.uk/~shamsbaa/Josephus.pdf Formulating The Extended Josephus Problem].</ref>
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book |author=[[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]] |title=[[Introduction to Algorithms]] |edition=Second Edition |publisher=MIT Press and McGraw-Hill |year=2001 |isbn=0-262-03293-7 |chapter=Chapter 14: Augmenting Data Structures |page=318}}
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| ==External links==
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| * [http://www.cut-the-knot.org/recurrence/flavius.shtml Josephus Flavius game] (Java Applet) at [[cut-the-knot]] allowing selection of every n<sup>th</sup> out of 50 (maximum).
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| * [http://mathworld.wolfram.com/JosephusProblem.html Josephus Problem at the MathWorld encyclopedia]
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| * [http://www.maa.org/publications/periodicals/loci/resources/the-josephus-problem Josephus Problem at Shippensburg University]
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| [[Category:Combinatorics]]
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| [[Category:Permutations]]
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| [[Category:Mathematical problems]]
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| [[Category:Computational problems]]
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A tenancy-in-widespread agreement is used whenever there may be widespread possession of a property. A person might go for actual property investment along with his siblings, buddies, or another get together. In any of those instances, it is a good option. It describes how the property is owned and what's the responsibility of each proprietor. That is the easiest way to divide the property with none problems.
b) A unit in an authorised condominium development. Take a look at our characteristic properties for a list of landed home for rent with photos and price. For a listing of serviced residences available, click on right here Investing within the property market is like playing the monopoly board recreation. It is advisable to have the proper strategies with the intention to achieve success. What higher technique to Resources">grasp the property investment strategies than by studying from the specialists and real property insiders. Factors that determine Singapore's property market A Look Back on 2011 and Wanting Forward to 2012 (at Propwise.sg) Each property has an AV. This AV of a property is estimated primarily based on market rentals of comparable or comparables properties. Tax Charge Previous to 1 Jan 2014 2.How is property tax calculated?
When you owner-occupy your own home, you'll be able to apply for owner-occupier tax charges If you're a Singaporean or Singapore Permanent Resident who personal and occupy just one private residential property or govt condominium, you need not apply because the concession will likely be granted routinely. It is easy to search out the best property agent to characterize you at HirePropertyAgent! Simply describe the kind of property, dimension, location and the expectations you need of the singapore property agent (eg. fee price, specialist in your space, experience, etc) Founder and chief executive of Second Likelihood Properties Mohamed Salleh Marican instructed The Straits Instances that each property class in Singapore appears to be coming off its peak price levels. Getting ready your property on the market
Its aroma acts as a stimulant. It is carminative, which implies it helps in relieving from gas. It is used to deal with many stomach-related problems. This natural drugs is used towards nausea, dyspepsia, flatulence, catarrh, rheumatism, and enteritis. It additionally has tonic and antibacterial properties. As a consequence of these two properties it's used in veterinary and homeopathic medicine. Its root can be utilized for treating fungal infections and in pain reduction of joints and muscles.
Goodrich stated ninety nine % of his shoppers buy for investment and so they search for yields of around 3 to 4 p.c. The highest 1 percent are on the lookout for a trophy property. Property management includes widespread prices and utilities. Guidelines concerning the precise usage of the property by the co-house owners together with issues like pets, noise levels, ground covering, and so forth., and responsibility of every co-proprietor for property upkeep. Other than the aforementioned factors, it additionally includes particulars concerning dispute decision, mortgage of the property, sale of property, proper to lease, enhancements and repairs, and termination and signatures. Within the Rest of Central Region , property costs were up by 0.9%. Outdoors Central Area , property costs rose by 3.8% q-q. 14xxpsf view this mission
But currently they have been transferring up market, brokers say. The current in-vogue building among the many Chinese language is Central Park's One57, a brand new skyscraper designed by Pritzker Prize-profitable French architect Christian de Portzamparc, the place they will spend $18.eighty five million for a 3-bedroom or $fifty five million for an house taking on your complete 81st floor. The constructing comes with all of the amenities of a 5-star resort. Shanghai businessman Wang Jiguang has already picked up two houses in one other major Australian metropolis, Melbourne, and one condo in Sydney. "My youngster is going to check abroad, and we are simply getting ready some abroad belongings for our baby, which will probably be much less risky," Wang said in a telephone interview from Shanghai. Uses in Hair and Skin Care
For properties owned by more than one owner, all house owners are collectively liable for paying property tax. For correspondence purpose, we are going to normally tackle to the owner who is listed first within the Notice of Transfer filed by the vendor's lawyer. Therefore, it's important that you as the new owner or one of the new homeowners, inform the seller's lawyer the one that will be responsible for the property tax issues. As soon as the switch record is up to date, we will likely be corresponding with the individual on all property tax issues, including payment of property tax.