The UK Linguistics Olympiad 2017

Transcription

1 Round 2 Prolem 5. Magi Yup ik Central Alaskan Yup ik elongs to the Eskimo-Aleut language family. It is spoken in western and southwestern Alaska y around 20,000 speakers. Yup ik people have an interesting approah to ounting: the words for the numers an e roken down into meaningful parts whih may e related to their ody parts. For example, the word for five, talliman, means one arm and the word for six, arvinlegen, means ross over, as you need to hange hand to go on ounting. Another interesting fat aout the Yup ik people is that their parkas often inlude geometrial patterns suh as the 3x3 square shown elow. a This prolem is aout an imaginary magi square ased on this pattern. In a magi square with nine ells, like this one, every ell ontains one of the digits 1 to 9, every one of these digits is found in just one ell, and the sum of all the digits in every row, olumn, and diagonal is the same. In our magi square, the middle ell of the top row (i.e. ell 2a) ontains the digit 9. The lues in the next tale will help you fill in the rest of the ells y defining eah row and eah olumn as a three-digit numer whih ould also e defined in words as a omplex numer-phrase. For example the sequene of digits defines the numer 123, whih an also e given in English y the numer-phrase One hundred and twenty three. The only snag is that the numer-phrases in the tale are in Yup ik. HINT: The Yup'ik name for the numer 294 is yuinaat qula etaman qula etaman. Aross Down a Yuinaat yuinaq etaman qula malruk 1 Yuinaat yuinaq atauiq akimiaq pingayun Yuinaat akimiaq malruk akimiaq malruk Yuinaat yuinaak malruk akimiaq atauiq 2 Yuinaat yuinaak malruk yuinaat malrunglegen qula atauiq 3 Yuinaat qula pingayun akimiaq atauiq Q.5.1. Fill in the magi square on your answer sheet. Q.5.2. Give the Yup ik numer-phrase for the numer defined y the diagonal of ells 1a-2-3. Q.5.3. Explain your answers.

2 Prolem 2.5 Magi Yup ik (answer lank) Q.5.1 a 9 Q.5.2 1a = Q.5.3 Explanation (if you need more paper, ask for it don t write on the ak of this sheet!)

3 Solution and marking. Soring (max 31) : 2 points for every orretly-filled ell. (max 16) 5.2: 1 point for every orret Yup ik word. (max 5) 5.3: (max 10) o 3 points for a mathematial explanation of the magi square. o 5 points for a linguisti explanation of the Yup ik numer system. o 2 points for an explanation of how the lue numers lead to the solution. a Yuinaat yuinaq malruk akimiaq atauiq. 5.3 Explanation: see the ommentary.

4 Commentary (y the prolem s author, Kai Low Rui Hao) Candidates should reognize this as a mathematis prolem and take into onsideration the possiility of the spelling of numers eing represented in a different ase (for example, in Frenh, 99 is quarte-vingt dix neuf, whih translates to 4 x ). There is only one possile type of 3-y-3 magi square, although there are 8 distint onfigurations that stems from the one type due to rotation and refletion. These permutations are avoided due to the presene of the Yup ik ross-numer puzzle lues. Sine there are 9 digits, the numers an e arranged via this formula N-1 N+4 N-3 N-2 N N+2 N+3 N-4 N+1 This formula takes into onsideration 9 onseutive numers, as well as the fat that the sum of every row, olumns, and diagonals is the same 3n. Sine the sum of eah olumn is 3n, the sum of three olumns is 9n. The sum from 1 to 9 is 45. Hene 9n = 45, leaving us with n=5. The answer to Task 1 is as aove. Candidates may in the proess arrive at the other 7 possile ominations. These permutations will e eliminated when heked against the yup ik hints. Sine the question provided 2Down as a 3-digit numer that starts with 9, and all numers in the lues are 3-digit, and all Yup ik numers egin with Yuinaat, andidates have to onsider that Yup ik may e using a ase larger than 10. The most ommon, in fat, is the Vigesimal system, whih is ase 20. This is also pratial ased on the akground provided in the question Yup ik people ased their onept of ounting on ody parts (20 fingers + toes). Despite the various permutations, andidates should e ale to arrive at 2D, whih is 951. Further attempts at solving the spelling will reveal that 951 = (20 x 20 x 2) + (20 x 7) + (10 + 1). This is true to the ase 20 system. Arriving at this onlusion will reveal: 1. Suffix k: multiply y 2/doule/to do with two (a dual numer) 2. Suffix t: multiply y more than 2 (a plural numer) 3. Suffix q: the root suffix. If the word ends in suffix q, it signifies to the andidate that the susequent numer should e an addition and not a multipliation. Candidates an then work on 3Aross, whih is the next iggest numer with 20 x 20 x 2. Hene, the numer, whih is the seond iggest after one starting with 9, should start with 8. This will lead to 3A eing 816. Candidates an then assoiate akimiaq atauiq to 16. From the 2D, atauiq = 1, hene akimiaq is 15. This is a reasonale and valid guess sine Yup ik people pay attentions to numers ased on the hands and feet.

5 Next, refer to 3D where andidates an make out (20 x (10+?)) Given that diagonals also sum up to 15, we an gather 2 and then 7 from it eing the remaining ell in 3D. Hene 3D is 276 and it is (20 x (10+3)) Knowing that qula = 10, akimiaq = 15, and that malruk = 2, pingayun = 3, we will e ale to solve most of the magi square. The other numer not mentioned is etaman =4. Task 2: Yuinaat yuinaq malruk akimiaq atauiq. The diagonal 1a-2-3 is 456 and an e expressed in Yup ik spelling as (20 x 20) + (20 x 2) However, notie from other lues that numers elow 800 are not phrased in this manner ut more of (20 x 22) Hene the numer will e expressed as (20 x (20 + 2)) + 16.