ÿþ<html> <head> <title>MM of Everything: How many squares are there on a chess board?</title> <style><!--a:hover{color:FF2222; }--></style> </head> <body bgcolor="#99FFFF" text = "#220088" link="#008000" alink="#008000" vlink="#008000" background="../../games/queens/chessbk.gif" > <meta name="keywords" content="Kjartan, Poskitt, Kjartan Poskitt, Murderous Maths, Murderous maths of everything, chess, chess boards, squares, pyramid numbers"> <meta name="description" content="How many squares are there on a chess board?"> <a href="../MMoE.htm"> <IMG SRC="MMoEcovsm.jpg" align=right border=0></A> <TABLE><TR><TD valign=top width=450><H1> <a href="../../indextxt.htm"> <img src="../../mmlogexp.gif" align=LEFT hspace=20 border=0></A> <font color ="#ee0000">How many squares are there on a chess board?</FONT></H1> <H4>Oh dear - this simple question leads to some big arguments between the chess pieces! <P>The board has 8 squares along the bottom and 8 squares up the side, so most people would say the answer is 8x8 =64. <BR>BUT suppose you count the BIG square as one, and then you also want to count up all the smaller squares - how many squares are there in total? </TD><TD> <img src="chess pix/pieces.gif"> </TD></TR></TABLE> <TABLE align=right width=225><TR><TD> <img src="../../games/queens/queenlogo.gif" align = left > If you like playing with chess boards, try <a href="../../games/queens/index.htm">The EIGHT QUEENS Puzzle!</a> </TD></TR></TABLE> <TABLE><TR><TD width=150 valign=top> <font size=10><B>1</b></font> <P>First of all the whole board makes one BIG square. </TD> <TD valign=top><img src="chess pix/chessnossm1.gif"></TD> </TR></TABLE> <P> <TABLE><TR><TD width=150 valign=top> <font size=10><B>+ 4</b></font> <P>Then there are four slightly smaller squares (measuring 7x7). </TD> <TD valign=top><img src="chess pix/chessnossm4.gif"></TD> <TD width=30></TD> <TD width=150 valign=top>We've marked where the corners go on the main board using numbers 1-4. <P>We've outlined where the first square goes in <font color="DD0000">red</font>. </TD> <TD valign=top><img src="chess pix/chessnos4.gif"> </TD> </TR></TABLE> <P> <TABLE><TR><TD width=150 valign=top> <font size=10><B>+ 9</b></font> <P>Then there are nine squares measuring 6x6. </TD> <TD valign=top><img src="chess pix/chessnossm9.gif"></TD> <TD width=30></TD> <TD width=150 valign=top>We've marked where the corners go on the main board using numbers 1-9. <P>We've outlined where square number 2 goes in <font color="4477FF">blue</font>. </TD> <TD valign=top><img src="chess pix/chessnossm9a.gif"> </TD> </TR></TABLE> <TABLE align=right border=1 bgcolor="FF0000" width=371><TR><TD bgcolor="000000"> <font color="FFFFFF"><Center> <P>When you add a list of square numbers you get the <BR><font size=5><B>Square Pyramid Numbers</b></font>. <P><img src="chess pix/sqpyrnos.gif"> <P>The formula for the number of balls in a square pyramid = <H3> [ 2<i>n</i><sup>3</sup> + 3<i>n</i><sup>2</sup> + <i>n</i> ] &#247; 6 </H3> where <i>n</i> = the number of layers of the pyramid. </center> <P> <Table align=right width=30><TR><TD><a href="../BKMM11.htm"> <img src="../pix/mm11covsm.gif"></A> </TD></TR></TABLE> The number of squares on a chessboard = the 8th pyramid number. So if you make <i>n</i>=8 in the formula you'll get the answer! <P> <color="FFFF00">Find more fantastic formulas in <a href="../BKMM11.htm">THE PERFECT SAUSAGE</A>. You can even find out how many <i>rectangles</i> are on a chess board using triangle numbers! <BR>(The answer is T<sub>8</sub> x T<sub>8</sub> = 36 x 36 = 1296). </TD></TR></TABLE> <P>So how many squares of different sizes have we counted so far? It's <font size=5><B>1 + 4 + 9 + ... </b></font>. <P>You might have noticed these are the square numbers. <font size=5><B> 1<sup>2</sup>=1 2<sup>2</sup>=4 3<sup>2</sup>=9 </b></font> <P> <TABLE><TR><TD width=350 valign=top> <P>Here's the next smaller size of square. How many can we fit on a chessboard? <P>It's the next square number! <BR>4<sup>2</sup> = <font size=10><B>+ 16</b></font> </TD> <TD valign=top><img src="chess pix/chessnossm16.gif"></TD> </TR></TABLE> <P> <TABLE><TR><TD width=150 valign=top> <P>Here are the other possible squares. </TD> <TD valign=top><Center><img src="chess pix/chessnossm25.gif"></TD> <TD width=30></TD> <TD valign=top><Center><img src="chess pix/chessnossm36.gif"></TD> <TD width=30></TD> <TD valign=top><Center><img src="chess pix/chessnossm49.gif"></TD> <TD width=30></TD> <TD valign=top><Center><img src="chess pix/chessnossm64.gif"></TD> </TR> <TR> <TD width=150>Numbers of each square that will fit on the board.</TD> <TD><Center><font size=8><B>+ 25</b></font></TD> <TD></TD> <TD><Center><font size=8><B>+ 36</b></font></TD> <TD></TD> <TD><Center><font size=8><B>+ 49</b></font></TD> <TD></TD> <TD><Center><font size=8><B>+ 64</b></font></TD> </TR></TABLE> <H3>So the total number of squares on the chess board = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = <font size=8><B>204</b></font> </H3> <P> <P><BR> <img src="mmlogonew2.gif" ALIGN=LEFT> <FONT FACE = "ARIAL"> <P><a href="../MMoE.htm" > The Murderous Maths of Everything</a> <P><a href="../../indextxt.htm" > Murderous Maths Home Page</a> </body> </html>