The,Hockey,Stick,Property,Pasc education The Hockey Stick Property of Pascal\\\'s Triangle
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The"Hockey Stick" property and the less well-known Parallelogramproperty are two characteristics of Pascal's triangle that are both intruigingbut relatively easy to prove. This article explains what these properties areand gives an explanation of why they will always work.The"Hockey Stick" property states that the sum of any diagonal linestarting from a 1 on the outside of the triangle is the number diagonally downfrom the last number, in a hockey stick shape. When the numbers of Pascal'striangle are left justified, this means that if you pick a number in Pascal'striangle and go one to the left and sum all numbers in that column up to thatnumber, you get your original number. This sounds very complicated, but it canbe explained more clearly by the example in the diagram below:1112 113 3 114 6 4115 1010 5 116 1520 15 6 117 2135 35 21 7 11+3+6+10+15+21= 35Trya couple of these sums out for yourself to get the hang of them. This is one ofmy favourite patterns in Pascal's triangle - it really it quite a surprisingthat this property seems to always work, and yet, as we are about to see, it isactually not too hard to prove!Asan example, I am going to shown the idea behind the proof with the sum shown inthe diagram above. We will start with the bottom of the Hockey Stick at 35, thetotal of the 1,3,6,10,15 and 21. As in Pascal's triangle every number is thesum of the two above it, we can start by writing the sum 35 = 15+20.Now,the 15 lies on the Hockey Stick line (the line of numbers in this case in thesecond column). But what can we do about the number 20? Change it into a sum ofthe two above! We get 20 = 10+10, and so our overall sum becomes 35 = 15+10+10. We now have a sum where both 15 and one of the 10s lie on the HockeyStick line. We continue this process, each time having only one number not onthe line, until we reach the edge of the triangle, where our number not on theline is a 1. Then, we are done because the remaining number we haven't got inour sum which is on the line is also a 1. The whole process for 35 is shownbelow (the numbers in boldare the ones which lie on the hockey stick line:35= 15+2035= 15+10+1035= 15+10+6+435= 15+10+6+3+1Itis clear, therefore, why the Hockey Stick property of Pascal's Triangle works,although this makes it no less an interesting pattern which can also bedeveloped into many other patterns such as the Parallelogram property. Normal 0 false false false EN-GB X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable{mso-style-name:"Table Normal";mso-tstyle-rowband-size:0;mso-tstyle-colband-size:0;mso-style-noshow:yes;mso-style-priority:99;mso-style-qformat:yes;mso-style-parent:"";mso-padding-alt:0cm 5.4pt 0cm 5.4pt;mso-para-margin-top:0cm;mso-para-margin-right:0cm;mso-para-margin-bottom:10.0pt;mso-para-margin-left:0cm;line-height:115%;mso-pagination:widow-orphan;font-size:11.0pt;font-family:"Calibri","sans-serif";mso-ascii-font-family:Calibri;mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:minor-fareast;mso-hansi-font-family:Calibri;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi;} Article Tags: Hockey Stick Property, Hockey Stick, Stick Property, Pascal's Triangle
The,Hockey,Stick,Property,Pasc