Task   (in three parts)



Part 1
Write a procedure (say 





c
o
9


(
x
)


{\displaystyle {\mathit {co9}}(x)}

) which implements Casting Out Nines as described by returning the checksum for 



x


{\displaystyle x}

. Demonstrate the procedure using the examples given there, or others you may consider lucky.

Part 2
Notwithstanding past Intel microcode errors, checking computer calculations like this would not be sensible. To find a computer use for your procedure:

 Consider the statement "318682 is 101558 + 217124 and squared is 101558217124" (see: Kaprekar numbers#Casting Out Nines (fast)).
 note that 



318682


{\displaystyle 318682}

 has the same checksum as (



101558
+
217124


{\displaystyle 101558+217124}

);
 note that 



101558217124


{\displaystyle 101558217124}

 has the same checksum as (



101558
+
217124


{\displaystyle 101558+217124}

) because for a Kaprekar they are made up of the same digits (sometimes with extra zeroes);
 note that this implies that for Kaprekar numbers the checksum of 



k


{\displaystyle k}

 equals the checksum of 




k

2




{\displaystyle k^{2}}

.
Demonstrate that your procedure can be used to generate or filter a range of numbers with the property 





c
o
9


(
k
)
=


c
o
9


(

k

2


)


{\displaystyle {\mathit {co9}}(k)={\mathit {co9}}(k^{2})}

 and show that this subset is a small proportion of the range and contains all the Kaprekar in the range.

Part 3
Considering this MathWorld page, produce a efficient algorithm based on the more mathematical treatment of Casting Out Nines, and realizing:

 





c
o
9


(
x
)


{\displaystyle {\mathit {co9}}(x)}

 is the residual of 



x


{\displaystyle x}

 mod 



9


{\displaystyle 9}

;
 the procedure can be extended to bases other than 9.
Demonstrate your algorithm by generating or filtering a range of numbers with the property 



k
%
(


B
a
s
e


−
1
)
==
(

k

2


)
%
(


B
a
s
e


−
1
)


{\displaystyle k\%({\mathit {Base}}-1)==(k^{2})\%({\mathit {Base}}-1)}

 and show that this subset is a small proportion of the range and contains all the Kaprekar in the range.



related tasks
 First perfect square in base N with N unique digits
 Kaprekar numbers


proc co9 {x} {
    while {[string length $x] > 1} {
	set x [tcl::mathop::+ {*}[split $x ""]]
    }
    return $x
}
# Extended to the general case
proc coBase {x {base 10}} {
    while {$x >= $base} {
	for {set digits {}} {$x} {set x [expr {$x / $base}]} {
	    lappend digits [expr {$x % $base}]
	}
	set x [tcl::mathop::+ {*}$digits]
    }
    return $x
}
 
# Simple helper
proc percent {part whole} {format "%.2f%%" [expr {($whole - $part) * 100.0 / $whole}]}
 
puts "In base 10..."
set satisfying {}
for {set i 1} {$i < 100} {incr i} {
    if {[co9 $i] == [co9 [expr {$i*$i}]]} {
	lappend satisfying $i
    }
}
puts $satisfying
puts "Trying [llength $satisfying] numbers instead of 99 numbers saves [percent [llength $satisfying] 99]"
 
puts "In base 16..."
set satisfying {}
for {set i 1} {$i < 256} {incr i} {
    if {[coBase $i 16] == [coBase [expr {$i*$i}] 16]} {
	lappend satisfying $i
    }
}
puts $satisfying
puts "Trying [llength $satisfying] numbers instead of 255 numbers saves [percent [llength $satisfying] 255]"