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### Solver for Sunday Times teaser 2021

by Brovnik (Hermit)
 on Jun 11, 2001 at 23:01 UTC Need Help??

```   1: #! /usr/local/bin/perl -w
2:
3: use strict;
4: use warnings;
5: use diagnostics;
6: use Data::Dumper;
7:
8: # I found the Sunday Times teaser 2021 interesting enough
9: # to write a program to solve after I had a few goes on
10: # paper and kept getting it wrong.
11: #
12: # It manages to do what should be a depth first search
13: # using a simple for loop due to the relative simplicity
14: # of the search space.
15: # P.S. There is a prize for sending in the first correct
16: # answers, but you'll need to have a copy of the ST in
17: # order to find the address.
18: #
19: # Problem taken from Sunday Times 10-Jun-01
20: #
22: # on a board with spaces numbered 0..N .
23: # A Player starts at 0, casts the die and moves his
24: # counter forward the number of squares shown. If he
25: # lands on a prime, he moves up to the next prime.
26: # If he lands on a square, he moves down to the next
27: # lower square (e.g. 25 to 16). Tom and I played a game.
28: # I went first, and we each threw the die seven times.
29: # Each time, after Tom threw, his position was double mine
30: # (and never zero).
31: #
32: # What were our positions after seven throws ?
33:
34: my \$last = 0;
35: my \$next;
36: my %primes = ();
37:
38: for (2,3,5,7,11,13,17,19,23,29,31,37,41,43,47)
39: {
40:    \$primes{\$last} = int(\$_);
41:    \$last = \$_;
42: }
43: delete \$primes{0};
44:
45: my %squares = ();
46: map {\$squares{\$_ * \$_} = int((\$_ - 1) * (\$_ - 1))} (1..10);
47:
48: my @posshim = ();
49: my @possme  = ();
50: my \$start   = 0;
51: my @carry   = ();
52: my @record  = (0);
53: my @all     = ();
54: my %record  = (0=>[0]);
55:
56: ### roll
57:
58: for (1..7)
59: {
60:    roll(\$_);
61: }
62: die "Failed, More than 1 possible sequence"
63:    if @all > 1;
64: die "Failed, no possible sequences"
65:    if @all < 0;
66:
67: print "-------\n\n";
68: report("My final Sequence",@{\$record{\$all[0]}});
69:
70:
71: #########
72:
73: sub roll
74: {
75:    my \$roll = shift;
76:    @carry = @record;
77:    @record = @all = ();
78:    report("-----\nRoll \$roll",@carry);
79:    for (@carry)
80:    {
81:       my \$start    = \$_;
82:       print "Starting with \$_\n------\n";
83:       @possme = MyTry(\$start);
84:       @posshim = HisTry(\$start*2);
85:
86:       report("His possibles        ",@posshim);
87:       #report("My  possibles",@possme);
88:
89:       my @newme  = carry_me(\@possme, @posshim);
90:       report("My Matching possibles",@newme);
91:       for my \$val (@newme)
92:       {
93:          push(@record,\$val) unless grep {\$_ == \$val} @record;
94:       }
95:       record(\$start,@newme);
96:    }
97: }
98:
99: sub record
100: {
101:    my \$end = shift;
102:    my @poss = @_;
103:    my \$i;
104:    my @ref;
105:    my \$found = -1;
106:
107:    #print Data::Dumper->Dump([\%record],["Record"]) , \$/;
108:    for my \$poss (@poss)
109:    {
110:       my @new = @{\$record{\$end}};
111:       push(@new,\$poss);
112:       \$record{\$poss} = \@new;
113:       report("Carrying Forward",@new);
114:       push(@all,\$poss);
115:
116:    }
117:    #print Data::Dumper->Dump([\%record],["Post - Record"]) , \$/;
118: }
119:
120: sub report
121: {
122:    my \$name = shift;
123:    my \$values = join(', ',@_);
124:    print "\$name => [\$values]\n";
125: }
126:
127: sub HisTry
128: {
129:    my \$start = shift;
130:    my @poss  = ();
131:    for my \$die (1..6)
132:    {
133:       my \$mid = \$start + \$die;
134:       my \$new = move(\$start,\$die);
135:       \$new = move(\$start,\$die);
136:       my \$valid = validHim(\$new);
137:       #print "Him => \$start + \$die => \$mid => \$new \$valid\n";
138:       push(@poss,\$new) if \$valid;
139:    }
140:    return @poss;
141: }
142:
143: sub MyTry
144: {
145:    my \$start = shift;
146:    my @poss  = ();
147:    for my \$die (1..6)
148:    {
149:       my \$mid = \$start + \$die;
150:       my \$new = move(\$start,\$die);
151:       \$new = move(\$start,\$die);
152:       my \$valid = validMe(\$new);
153:       #print "Me  => \$start + \$die => \$mid => \$new \$valid\n";
154:       push(@poss,\$new) if \$valid;
155:    }
156:    return @poss;
157: }
158:
159: sub carry_me
160: {
161:   my \$ref = shift;
162:   my @him = @_;
163:   my @new = ();
164:   for my \$value (@\$ref)
165:   {
166:      push(@new,\$value) if grep {\$value*2 == \$_} @him;
167:   }
168:   return @new;
169: }
170:
171: sub validHim
172: {
173:    my \$value = shift;
174:    return 0 if \$value % 2;
175:    return 0 if \$value == 0;
176:    return 1;
177: }
178:
179: sub validMe
180: {
181:    my \$value = shift;
182:    return 0 if \$value == 0;
183:    return 1;
184: }
185:
186: sub move
187: {
188:    my \$old = shift;
189:    my \$roll = shift;
190:    my \$new = \$old + \$roll;
191:    \$new = \$primes{\$new} if defined(\$primes{\$new});
192:    \$new = \$squares{\$new} if defined(\$squares{\$new});
193:    return int(\$new);
194: }
```

Replies are listed 'Best First'.
Re: Solver for Sunday Times teaser 2021
by CharlesClarkson (Curate) on Jun 22, 2001 at 07:47 UTC

Spladders, a game like snakes and ladders, is played on a board with spaces numbered 0..N. A Player starts at 0, casts the die and moves his counter forward the number of squares shown.

Is it a six-sided die?

The output looks like:
```-----
Roll 1 => [0]
Starting with 0
------
His possibles         => [6]
My Matching possibles => [3]
Carrying Forward => [0, 3]
-----
Roll 2 => [3]
Starting with 3
------
Is roll 1 a 0? Or am I missinterpreting the data?

Charles K. Clarkson
Hmmm..., looking at it again, I can see that it might be a bit confusing.

If you uncomment the print line in HisTry and MyTry, you get more verbose reporting about the results of each possibility.

```  Roll 1 => [0]
Starting with 0
This is where we start the game, with both players @ 0.
In later rounds, the round could start with several possibilities (e.g. Roll 7 => [14, 15]), and each is explored in turn.
```  ------
His possibles         => [6]
we try all 6 possibilities for the die in HisTry But, \$new = move(\$start,\$die) will adjust the output if it is a prime or a square before returning the value, and validHim(\$new) knocks out any of these that won't be valid final values.

So for the first roll :

```Him => 0 + 1 => 1 => 0 0
Him => 0 + 2 => 2 => 3 0
Him => 0 + 3 => 3 => 5 0
Him => 0 + 4 => 4 => 1 0
Him => 0 + 5 => 5 => 7 0
Him => 0 + 6 => 6 => 6 1
a   b    c    d e
a = start
b = die roll
c = start + die roll
d = adjusted down for square, up for prime.
e = is it a valid final number ?
Scores of 1,4 move down to the next square, and 2,3,5 move up to the next prime.

Since we have to end up with His score twice mine, his end number must be even, so this only leaves 6 from the first roll.

Do the same for Me, but matching possibilities are only those where his roll ends up as twice mine.

```  My Matching possibles => [3]
And carry forward to the next round the sequence of final values for Me.
In this case, I end up at the end of Roll 1 with a 3 as the only possible value that is valid under the contstraints.
```  Carrying Forward => [0, 3]
-----
Roll 2 => [3]
Starting with 3
.
.
.
My final Sequence => [0, 3, 4, 7, 10, 12, 14, 15]
The last line means that it found exactly one final set of rolls that would meet the constraints, and Me end up with 15, and Him on 30.

Note that several of the rolls carry forward multiple possible routes to the next round where they then get eliminated, and in Roll 5,6, we can get from 10=>12=>15 or 10=>13=>15 via 2 different routes but, since the answer is the same we don't care.

Hope that helps explain it.
You can also uncomment the Data::Dumper->Dump lines if you want to see the internal tables it uses build up during the game.
--
Brovnik

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