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\par This is an investigation of sequences formed by writing down the sum of
the squares of the digits of each number to get the next number in the
sequence. A number is called happy if it starts a sequence that goes to a
number which is repeated over and over again (e.g. 1, 1, 1, 1, \$\ldots\$) and this is
called a fixed point. Other numbers lead to cyclic patterns which repeat over
and over again. For example this sequence leads to a repeating 8-cycle: 

\par
25, 29, 85, 89, 145, 42, 20, 4, 16, 37, 58, 89, \$\ldots\$

\par
Rachel Walker from the Mount School, York explained that in base ten, 122 is
a sad number because it has an 8-cycle like 25 and this is her solution: 
\par
122, 9, 81, 65, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58........

\par
You can find a lot of 'happy numbers' such as: 310, 70, 86, 130, 7, 5555,
1212, 13, 1000 and other numbers which are variations of these, e.g. swapping
the numbers the other way around (13, 31) and adding zeros onto the end of
numbers (31, 310). Also you get numbers that add up to other happy numbers
e.g. 5555 - four fives squared = 100.  

\par
The following sequences are the patterns of fixed points and cycles in base
eight that I have found.  These are fixed points so they are happy numbers
and any numbers which start sequences ending like this are happy numbers. 

\newline 64, 64, 62, 64, 64, \$\ldots\$
\newline 24, 24, 24, 24, 24, \$\ldots\$
\newline 1, 1, 1, 1, ,1, \$\ldots\$

\par
These are 2-cycles:
\newline 32, 15, 32, 15, 32, 15, \$\ldots\$
\newline 20, 4, 20, 4, 20, 4, \$\ldots\$

\par
This is a 3-cycle:
\newline 31, 12, 5, 31, 12, 5, 31, 12, 5, \$\ldots\$

\par
Pen Areecharoenlert, also from The Mount School, York found some more happy
numbers and cycles in base eight. Pen says \"There does not appear to be any
pattern in which ones are happy, but it looks as though there are 3 fixed
points: 1, 24 and 64.\" This is Pen's list. 

\par
\newline \textbf{1, 1, 1, \$\ldots\$ so 1 is happy.}
\newline 2, 4, 20, 4, 20, \$\ldots\$ so 2 is sad. (2-cycle)
\newline 3, 11, 2, 4, 20, \$\ldots\$ so 3 is sad. (2-cycle)
\newline 4 is sad. (2-cycle)
\newline 5, 31, 12, 5, \$\ldots\$ so 5 is sad. (3-cycle)
\newline 6, 44, 40, 20, 4, 20, \$\ldots\$ so 6 is sad. (2-cycle)
\newline 7, 61, 45, 51, 32, 15, 32, \$\ldots\$ so 7 is sad. (2-cycle)
\newline \textbf{10,  1,  1,  1,  1\$\ldots\$ so 10 is happy.}         
\newline 11,  2,  4,  20,  4, \$\ldots\$  so 11 is sad. (2-cycle)
\newline 12,  5,  31,  12, \$\ldots\$ so 12 is sad. (3-cycle)  
\newline 13, 12, \$\ldots\$ so 13 is sad. (3-cycle)
\newline 14,  21, 5, \$\ldots\$ so 14 is sad. (3-cycle)
\newline 15,  32,  15,\$\ldots\$ so 15 is sad. (2-cycle)
\newline 16,  45,  51, 32, \$\ldots\$ so 16 is sad. (2-cycle)
\newline 17, 62, 50, 31,\$\ldots\$ so 17 is sad. (3-cycle)
\newline 20,  4,  20,  4, \$\ldots\$ so 20 is sad. (2-cycle)
\newline 21,  5, \$\ldots\$  so 21 is sad. (3-cycle)
\newline \textbf{22, 10, 1, 1,\$\ldots\$ so 22 is happy.}
\newline 23, 15, \$\ldots\$ so 23 is sad. (2-cycle)
\newline \textbf{24,  24,  24,  24, so 24 is happy.}
\newline \textbf{25,  35,, 42, 24, 24,\$\ldots\$ so 25 is happy.}
\newline 26, 50, 31,\$\ldots\$  so 26 is sad. (3-cycle)
\newline 27,  65,  75, 112, 6, 44, 40, 20,\$\ldots\$  so 27 is sad. (2-cycle)
\newline 30 , 11,  2,  4,  20, \$\ldots\$  so 30 is sad. (2-cycle)
\newline 31,  12, \$\ldots\$  so 31 is sad. (3-cycle)  
\newline 32, 15, \$\ldots\$ so 32 is sad. (2-cycle)
\newline \textbf{33,  22,  10,  1, \$\ldots\$ so 33 is happy.}
\newline 34, 31, 12, \$\ldots\$ so 34 is sad. (3-cycle)
\newline \textbf{35,  42,  24,  24, \$\ldots\$ so 35 is happy.} 
\newline 36,  55,  62, 50, 31, \$\ldots\$  so 36 is sad. (3-cycle)  
\newline 37, 72,  65, 75, 112 so 37 is sad. (2-cycle)
\newline 40 > 20 > 4 > 20 so 40 is sad. . (2-cycle)
\newline 41,  21, 5, 31, 12, 5, \$\ldots\$ so 41 is sad. (3-cycle)
\newline \textbf{42,  24, 24,\$\ldots\$ so 42 is happy.}
\newline 43,  31, \$\ldots\$ so 43 is sad. (3-cycle)
\newline 44,  40, 20, 4, 20,\$\ldots\$ so 44 is sad. (2-cycle)
\newline 45, 51, 32, \$\ldots\$ so 45 is sad. (2-cycle)
\newline \textbf{46,  64,  64, 64, \$\ldots\$ so 46 is happy.}
\newline 47, 101, 2,  4,  20, \$\ldots\$ so 47 is sad. (2-cycle)

\par
Claire Kruithof, Madras College, St Andrews proved that 24 and 64 are the
only 2 digit fixed points in base 8. This is Claire's proof: 

\par
\"We wanted to see if there would be any more fixed points so we used
algebra. The number following \$ab\$ is \$a^2 + b^2\$ where \$a\$ and \$b\$ are
between 0 and 7.  For a fixed point we have: 

\par
\[a^2 + b^2 = 8a + b\]
\[b^2 - b = 8a - a^2\]
\[b(b - 1) = a (8 - a)\]

\par
Searching for possibilities 
" -->

    
<table border="1">
<tr>
<td>B</td>
<td>a</td>
</tr>
<tr>
<td>1 x 0 = 0</td>
<td>7 x 1 = 7</td>
</tr>
<tr>
<td>2 x 1 = 2</td>
<td>6 x 2 = 12</td>
</tr>
<tr>
<td>3 x 2 = 6</td>
<td>5 x 3 = 15</td>
</tr>
<tr>
<td>4 x 3 = 12</td>
<td>4 x 4 = 16</td>
</tr>
<tr>
<td>5 x 4 = 20</td>
<td></td>
</tr>
<tr>
<td>6 x 5 = 30</td>
<td></td>
</tr>
<tr>
<td>7 x 6 = 42</td>
<td></td>
</tr>
</table>
<br />
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\par
The only possibilities are \$a = 2, \ b = 4\$ and \$a = 6, \ b = 4\$. 

\par
With more digits the sums of squares are never bigger than some multiple of
\$7^2\$. It can be proved that sequences cannot go on for ever without
repeating patterns and so all sequences go into cycles or go to a fixed
point. Moreover it can be proved that there are no fixed points with 3 or
more digits. 


" -->