PROPERTIES OF AN EULER SQUARE

Transcription

1 PROPERTIES OF N EULER SQURE bout 0 the mathematicia Leoard Euler discussed the properties a x array of letters or itegers ow kow as a Euler or Graeco-Lati Square Such squares have the property that every row ad colum cotais each elemet just oce ad that all rows ad colums add up to (+)/ whe usig the first itegers Here is a elemetary example- Notice that the sum of the elemets equals exactly (+)= The simplest way to costruct such a square is to start with a sigle row cotaiig all elemets i radom order Next oe shifts a idetical secod row to the left by oe space ad cotiues this procedure dow to the last row cotiuig the shiftig of oe space to the left for each row shift The result is a Euler Square Here is a example of a costructed x square whe the first row reads [ ]- You will ote each row ad colum cotai all four itegers just oce ad the sum of the elemets i ay row or colum is exactly (+)/=0 The sum of the elemets i the two diagoals will geerally ot be equal y iterchagig rows ad/or colums i this Euler Square, oe ca create umerous other possibilities such as the followig - ad

2 This shows that there are multiple possibilities for ay x square but oly oe uique form whe the first row is represeted by the ascedig values [,,,, ] We call this special form a stadard Euler Square Its form is give by the matrix- M Notice this form satisfies all the properties for a stadard x Euler Square icludig that the sum i ay row or colum equals precisely- S ( ) ( ) For eve this square matrix M has the trace(sum of the elemets alog the mai diagoal)- Tr ( M ) / k (k ) For odd the trace has the still simpler form- Tr ( M ) S ( ) so that it equals the colum ad row sums The sum of the elemets alog the right secodary diagoal is always regardless of whether is eve or odd elow you will fid a table summarizig the properties of differet sized stadard Euler Squares startig with = through =9- Trace bsolute Value of Determiat

3 oth the trace ad the absolute value of the determiat det(m) icrease with icreasig with the rapid determiat growth becomig especially oticeable The Euler Squares ca be thought of as precursors to certai mathematical puzzles such as Sudoku Ideed it will usually be sufficiet to costruct a full (but ot ecessarily stadard) x square by just startig with a set of o-repeatig itegers osider the very simple case of a x Euler Square- where the dashes represet the locatio of the remaiig six ukows We kow that each row ad colum must add up to So oe sees at oce that elemet a, = ad a, = These results i tur force a, =, a, =, a, = ad a, =, producig the completed square- We could also have produced this square startig with elemets,, ad i the first row ad the movig thigs oe uit to the right i row two, ad a additioal uit to the

4 right i row three Sice this is ot a stadard Euler Square, the above trace law will o loger hold lso the iitial placemet of the kow elemets caot be radom but will allow a solutio oly whe placed i a certai patter other iterestig property of Euler squares becomes clear by lookig at the square formed by the product of two sets [,, ] ad [,, ] - I this square we otice that both- ad already represet stadard Euler Squares Their product also is Take the case of =, =, ad = This produces a Euler Square- It also works whe the elemets are added together such that- Notice that i both cases the sum of the rows ad colums have the ew value of +++++=

5 To see if this procedure works for larger squares, we start with the case- D D D D d let =, =, =, ad D= y first multiplyig the elemets together, we retrieve the Euler Square We ote however if D had bee or some other umbers, the procedure would ot have worked for the the first row would have cotaied the same elemet twice ddig elemets for the values of,,, ad D give will ot work So we ca state that Geeratig ew x squares from two sets of elemets which idividually lead to Euler Squares will work whe multiplyig or addig the elemets together oly if oe of the resultat elemets repeat themselves i ay give row or colum If we took =, =, =, ad D= thigs would work fie, producig ad 9 I this case the sum of the rows ad colums would be 0 ad 0, respectively There is othig restrictig the elemets of the matrix represetig Euler Squares cotaiig more tha two compoets Thus oe has the x stadard Euler Square-

6 DEF GHI DEF GHI GHI DEF From this oe also has the squares- D G E H F I D G E H ad F I G D H E I F s log as the elemets i a give row or colum ot equal each other, the squares will be stadard Euler Squares For example, [,,,D,E,F,G,H,I]=[,,,,,,,,9] will work Fially we ask if the Euler squares ecoutered above ca take o a more symmetric form The aswer is i the affirmative for x squares provided is eve Such squares ( which we will refer to as Sym-Squares) are easiest to costruct graphically as show i the followig figure for a x square- Oe starts with a large square box ad divides it ito = = equal size smaller boxes which are i tur broke ito four equal sized right triagles by the crossig of two diagoal lies Numbers through are the placed at the crossig poits of the outer twelve boxes i the

7 maer show The square is completed by placig the umbers ad at the crossigs i the remaiig four ier boxes The result is a Sym-Square The umbers alog the left diagoal of the origial box are all oes ad alog the right diagoal are fours colorized versio of a x Sym-Square follows- If oe ow goes to a x Sym-Square, its matrix form will have a trace of with all elemets alog this diagoal beig log the right diagoal each elemet will have the value so that the total sum is Here is the result for a x Sym-Square for which =- The elemets are easily recogized oce the mai diagoals ad the values i the outer two rows ad colums have bee recorded Note the perfect symmetry of this square about the mai (left) diagoal alog which all the elemets are oe Here the sum of the elemets alog ay row or colum equals (9)/= ad is thus the same as for the correspodig stadard x Euler Square colorized versio of this square looks like this-

8 Here there are eight small squares for each of the eight colors used The symmetries are easy to recogize from the color patter Each row ad colum cotais a give color oly oce March 0