From 5f65d62528e942d20397e63c2adeb3bbbb380f5c Mon Sep 17 00:00:00 2001
From: Jörg Sommrey <28217714+jo-37@users.noreply.github.com>
Date: Tue, 18 May 2021 09:22:14 +0200
Subject: Solution to task 1

---
 challenge-113/jo-37/perl/ch-1.pl | 92 ++++++++++++++++++++++++++++++++++++++++
 1 file changed, 92 insertions(+)
 create mode 100755 challenge-113/jo-37/perl/ch-1.pl

diff --git a/challenge-113/jo-37/perl/ch-1.pl b/challenge-113/jo-37/perl/ch-1.pl
new file mode 100755
index 0000000000..2b9be921d4
--- /dev/null
+++ b/challenge-113/jo-37/perl/ch-1.pl
@@ -0,0 +1,92 @@
+#!/usr/bin/perl -s
+
+use v5.16;
+use Test2::V0;
+use experimental qw(signatures postderef);
+
+our $examples;
+
+run_tests() if $examples;	# does not return
+
+die <<EOS unless @ARGV;
+usage: $0 [-examples] [N D]
+
+-examples
+    run the examples from the challenge
+ 
+N
+    number to be broken down into summands
+
+D
+    digit to appear in summands
+
+EOS
+
+
+### Input and Output
+
+my ($n, $d) = @ARGV;
+say rep_int($n, $d);
+
+
+### Implementation
+
+# Some considerations:
+# - The task neither requires the summands to be distinct nor a solution
+#   to have more than one summand.
+# - Every integer n that is a multiple of the digit d can be represented
+#   as a multiple sum of the given digit.
+# - Every integer having d in its decimal representation is a solution
+#   with itself as the sole summand.
+# - For all d > 0 and 10 * d <= n < 10 * (d + 1) the number starts
+#   with the digit d and thus is a solution itself.
+# - For all d > 0 and 10 * (d + 1) <= n there is a number m with
+#   10 * d <= m < 10 * (d + 1) starting with d and n - m is a multiple
+#   of d.  Thus n is representable as a sum of numbers that have the
+#   digit d in their decimal representation.
+# - For d = 0 and 100 <= n an analogous consideration is applicable when
+#   taking d=10 instead.  As leading zeros do not count, with the taken
+#   modification the second digit becomes zero.
+# - The remaining cases are n < 10 * d with the modified d.  Further
+#   analysis can be applied to these, e.g checking the special cases
+#   where d is one, even or five or is already occurring in n.  However,
+#   skipping any refinements and performing a brute force approach on
+#   this small solution space instead.
+
+sub rep_int ($n, $d) {
+    $d ||= 10;
+    return 1 if $n >= $d * 10;
+
+    # keys are strings, using the numeric values.
+    my %sum = (0 => 0);
+
+    # All numbers containing the digit $d.
+    for (my $num = $d; $num <= $n; $num += 10) {
+        # All sums found so far.
+        for my $sum (values %sum) {
+            # New sums arise from the current sum plus multiples of the
+            # current number.
+            for (my $new = $sum + $num; $new <= $n; $new += $num) {
+                return 1 if $new == $n;
+                $sum{$new} = $new;
+            }
+        }
+    }
+
+    # Not found.
+    0;
+}
+
+
+
+
+### Examples and tests
+
+sub run_tests {
+
+    is rep_int(25, 7), F(), 'example 1';
+    is rep_int(24, 7), T(), 'example 2';
+
+    done_testing;
+    exit;
+}
-- 
cgit