Long stairs
An evil wizard has trapped you in the stairwell of his castle. Your only hope of escape is to run up all 100 steps to reach the top, and you can run one step per second. Unfortunately the wizard uses magic to lengthen the staircase by five steps per second. The new steps are inserted randomly between existing steps, so if you're lucky some of them might be beneath you and not ahead of you.
Can you escape, or are you doomed to run up an ever-lengthening staircase forever?
Write a program to simulate your escape attempt. How are you doing after ten minutes? For every second between 600 and 609 seconds inclusive print the number of steps behind you and the number still ahead of you.
If you escaped, run 10,000 tests and print the average time taken and the average final length of the staircase.
C
<lang c>
- include <stdio.h>
- include <stdlib.h>
- include <time.h>
int main(void) {
int trial, secs_tot=0, steps_tot=0; //keep track of time and steps over all the trials
int sbeh, slen, wiz, secs; //all the variables related to an individual trial
time_t t;
srand((unsigned) time(&t)); //random number seed
printf( "Seconds steps behind steps ahead\n" );
for( trial=1;trial<=10000;trial++ ) { //10000 attempts for the runner
sbeh = 0; slen = 100; secs = 0; // initialise this trial
while(sbeh<slen) { //as long as the runner is still on the stairs
sbeh+=1; //runner climbs a step
for(wiz=1;wiz<=5;wiz++) { //evil wizard conjures five new steps
if(rand()%slen < sbeh)
sbeh+=1; //maybe a new step is behind us
slen+=1; //either way, the staircase is longer
}
secs+=1; //one second has passed
if(trial==1&&599<secs&&secs<610)
printf("%d %d %d\n", secs, sbeh, slen-sbeh );
}
secs_tot+=secs;
steps_tot+=slen;
}
printf( "Average secs taken: %f\n", secs_tot/10000.0 );
printf( "Average final length of staircase: %f\n", steps_tot/10000.0 );
return 0;
}</lang>
- Output:
Seconds steps behind steps ahead 600 2147 953 601 2151 954 602 2156 954 603 2160 955 604 2166 954 605 2169 956 606 2174 956 607 2180 955 608 2184 956 609 2188 957 Average secs taken: 2921.493200 Average final length of staircase: 14707.466000
Factor
<lang factor>USING: combinators.random io kernel math math.order prettyprint ;
- position ( -- n ) 0 ;
- stairs ( -- n ) 100 ;
- new ( -- m n ) position stairs ;
- incd ( m n -- o p ) swap 1 + swap ;
- seconds ( m -- n ) 100 - 5 / ;
- window? ( n -- ? ) seconds 600 609 between? ;
- zap ( m n -- o p ) 2dup / [ incd ] whenp 1 + ;
- barrage ( m n -- o p ) 5 [ zap ] times ;
- .n ( n -- ) pprint 7 [ bl ] times ;
- .stats ( m n -- ) dup seconds .n over .n swap - .n nl ;
- .header ( -- ) "Seconds behind ahead" print ;
- ?.status ( m n -- o p ) dup window? [ 2dup .stats ] when ;
- tick ( m n -- o p ) incd barrage ;
- demo ( m n -- o p ) tick ?.status ;
- sim-with ( q -- n ) new rot [ 2dup < ] swap while drop ; inline
- first ( -- n ) [ demo ] sim-with ;
- sim ( -- n ) [ tick ] sim-with ;
- sims ( -- n ) 0 first + 9999 [ sim + ] times ;
- steps ( m -- n ) "Avg. steps: " write 10000 / dup . ;
- time ( n -- ) "Avg. seconds: " write seconds . ;
- main ( -- ) .header sims nl steps time ;
main</lang>
- Output:
Seconds behind ahead 600 2067 1033 601 2073 1032 602 2078 1032 603 2082 1033 604 2087 1033 605 2091 1034 606 2096 1034 607 2099 1036 608 2103 1037 609 2108 1037 Avg. steps: 14731+23/200 Avg. seconds: 2926+223/1000
And for fun, a version without stack effects. <lang factor>USING: combinators.random effects.parser io kernel math math.order parser prettyprint stack-checker words ;
<< SYNTAX: .: [ scan-new-word parse-definition dup infer ]
with-definition define-declared ; >>
.: position 0 ; .: stairs 100 ; .: new position stairs ; .: incd swap 1 + swap ; .: seconds 100 - 5 / ; .: window? seconds 600 609 between? ; .: zap 2dup / [ incd ] whenp 1 + ; .: barrage 5 [ zap ] times ; .: .n pprint 7 [ bl ] times ; .: .stats dup seconds .n over .n swap - .n nl ; .: .header "Seconds behind ahead" print ; .: ?.status dup window? [ 2dup .stats ] when ; .: tick incd barrage ; .: demo tick ?.status ; .: first new [ 2dup < ] [ demo ] while drop ; .: sim new [ 2dup < ] [ tick ] while drop ; .: sims 0 first + 9999 [ sim + ] times ; .: steps "Avg. steps: " write 10000 / dup . ; .: time "Avg. seconds: " write seconds . ; .: main .header sims nl steps time ;
main</lang>
Fermat
<lang fermat> time_tot:=0; {we'll keep track of the total time and steps across all the trials} steps_tot:=0; !!('Seconds ','Steps behind ','Steps ahead'); for trial = 1 to 10 do
s_behind:=0; {initialise staircase and runner for this trial}
stair_len:=100;
time:=0;
while s_behind < stair_len do
s_behind:=s_behind + 1;
for wiz = 1 to 5 do
stair_len:=stair_len + 1; {evil wizard creates a step} {(mwaahahaha)}
if (Rand|stair_len) < s_behind then
s_behind:=s_behind + 1;
fi
od;
time:=time+1; {another second has passed}
if trial = 1 and 599<time and time<610 then
!!(time, ' ', s_behind, ' ',stair_len-s_behind)
fi;
od;
time_tot:=time_tot+time;
steps_tot:=steps_tot+stair_len;
od;
!!(time_tot/10000); !!(steps_tot/10000);</lang>
- Output:
Seconds Steps behind Steps ahead
600 2125 975 601 2129 976 602 2132 978 603 2136 979 604 2142 978 605 2146 979 606 2151 979 607 2156 979 608 2162 978 609 2166 979 3697083 / 1250 3722083 / 250
FreeBASIC
<lang freebasic> randomize timer dim as uinteger steps_behind = 0, stairs_length = 100, seconds, j dim as uinteger seconds_tot, steps_tot print "Seconds", "steps behind", "steps ahead" for trial as uinteger = 1 to 10000 'We'll have the runner try this 10000 times
steps_behind = 0 'runner starts at the bottom
seconds = 0 'reset time taken
stairs_length = 100 'Staircase has 100 steps
while steps_behind < stairs_length 'if the runner hasn't reached the top
steps_behind += 1 'go up one step
for j = 1 to 5 'The evil wizard conjures another five steps
if int(rnd*stairs_length) < steps_behind then steps_behind += 1
'there's a chance that a new step will be behind you
stairs_length += 1 'but either way the staircase is one step longer
next j
seconds += 1 'that all took one second
if trial = 1 and seconds >599 and seconds < 610 then print seconds, steps_behind, stairs_length - steps_behind
'for the first attempt, see how the runner is doing after ten minutes
wend
seconds_tot += seconds 'if the runner escaped, track the time taken and the length of the stairs
steps_tot += stairs_length
next trial
print "Average time taken: ";seconds_tot/10000; " seconds." print "Average final staircase length: ";steps_tot/10000; " steps." 'if you noticed that last number is about 100*exp(5), that's no coincidence</lang>
- Output:
Seconds steps behind steps ahead 600 2032 1068 601 2038 1067 602 2042 1068 603 2045 1070 604 2048 1072 605 2053 1072 606 2055 1075 607 2060 1075 608 2064 1076 609 2068 1077 Average time taken: 2921.9457 seconds. Average final staircase length: 14709.7285 steps.
GW-BASIC
<lang gwbasic> 10 RANDOMIZE TIMER 20 TIMET = 0 : STEPST = 0 30 FOR TRIAL = 1 TO 10000 40 TIME = 0 50 SBEH = 0 60 SLEN = 100 70 SBEH = SBEH + 1 80 IF SBEH >= SLEN THEN GOTO 160 90 FOR WIZ = 1 TO 5 100 IF INT(RND*SLEN)<SBEH THEN SBEH = SBEH + 1 110 SLEN = SLEN + 1 120 NEXT WIZ 130 TIME = TIME + 1 140 IF TRIAL = 1 AND 599<TIME AND TIME <610 THEN PRINT TIME, SBEH, SLEN-SBEH 150 GOTO 70 160 TIMET = TIMET+TIME+1 170 STEPST = STEPST + SLEN 180 NEXT TRIAL 190 PRINT TIMET/10000 200 PRINT STEPST/10000</lang>
Julia
<lang julia>""" https://rosettacode.org/wiki/Long_stairs """
using Statistics
struct LongStairs
startstep::Int startlength::Int climbersteps::Int addsteps::Int
end
Base.length(LongStairs) = typemax(Int) Base.eltype(ls::LongStairs) = Vector{Int}
function Base.iterate(s::LongStairs, param = (s.startstep, s.startlength))
pos, len = param
pos += s.climbersteps
pos += sum(pos .> rand(1:len, s.addsteps))
len += s.addsteps
return [pos, len], (pos, len)
end
ls = LongStairs(1, 100, 1, 5)
println("Seconds Behind Ahead\n----------------------") for (secs, (pos, len)) in enumerate(collect(Iterators.take(ls, 609))[600:609])
println(secs + 599, " ", pos, " ", len - pos)
end
println("Ten thousand trials to top:") times, heights = Int[], Int[] for trial in 1:10_000
trialstairs = LongStairs(1, 100, 1, 5)
for (sec, (step, height)) in Iterators.enumerate(trialstairs)
if step >= height
push!(times, sec)
push!(heights, height)
break
end
end
end @show mean(times), mean(heights)
</lang>
- Output:
Seconds Behind Ahead ---------------------- 601 1938 1167 602 1944 1166 603 1948 1167 604 1952 1168 605 1956 1169 606 1959 1171 607 1963 1172 608 1967 1173 609 1971 1174 Ten thousand trials to top: (mean(times), mean(heights)) = (2927.0853, 14735.4265)
Pascal
Free Pascal
Trying to calculate results of multiple random rounds.Now a partial step forward 1/CntOfSpellStairs for every spell stair.
Now results are much closer.
<lang pascal>program WizardStaircase;
const
StartStairLength = 100; StairsPerSpell = 5; rounds = 10000;
procedure OutActual(trials,behind,infront:longword); begin
writeln(trials:5,infront+behind:12,behind:10,inFront:9);
end;
function OneRun(StairsPerSpell:Cardinal;WithOutput:boolean):longword; var
inFront,behind,total,i,trials: Cardinal;
begin
trials := 0;
total := StartStairLength;
inFront := total;
behind := 0;
repeat
inc(trials);
inc(behind);
dec(infront);
//spell
i := StairsPerSpell;
repeat
if random(total) < behind then
inc(behind)
else
inc(infront);
inc(total);
dec(i);
until i=0;
if WithOutput then
begin
if (trials <= 609)AND(trials>=600) then
OutActual(trials,behind,infront);
end;
until infront = 0;
OneRun := total;
end;
procedure CheckDouble(StairsPerSpell:Cardinal); var
behind,infront,relpos,total,One,relSpell : double; i : LongInt;
begin
write(StairsPerSpell:3);
One := 1.0;
behind := 0.0;
inFront := StartStairLength;
total := StartStairLength;
relSpell := One/StairsPerSpell;
repeat
i := StairsPerSpell;
//doing a partial move per one stair per spell
repeat
relpos := (behind+relSpell)/total;
behind := behind + relpos +relSpell;
inFront := inFront+(One-relpos)-relSpell;
total := total+One;
dec(i);
until i= 0;
until infront < One;
writeln((total-StartStairLength)/StairsPerSpell:10:0,total:14:0);
end;
var
i, mySpell, attempt, minStairs, maxStairs, total : longword;
begin
randomize;
writeln('Seconds steps total behind ahead');
total := OneRun(StairsPerSpell,true);
writeln(' average stairs needed steps minimum maximum');
for mySpell := 1 to 7 do
begin
write(mySpell:3,' ');
total := 0;
minStairs := High(minStairs);
maxStairs := low(maxStairs);
For i := rounds-1 downto 0 do
begin
attempt:= OneRun(mySpell,false);
if minStairs>attempt then
minstairs := attempt;
if maxStairs<attempt then
maxstairs := attempt;
inc(total,attempt);
end;
writeln((total-StartStairLength)/rounds/mySpell:12:3,total/rounds:14:3,minStairs:9,maxStairs:10);
end;
writeln;
writeln(' calculated ');
For i := 1 to 10 do
CheckDouble(i);
end.</lang>
- @ TIO.RUN:
Seconds steps total behind ahead
600 3100 2260 840
601 3105 2264 841
602 3110 2269 841
603 3115 2275 840
604 3120 2281 839
605 3125 2285 840
606 3130 2291 839
607 3135 2295 840
608 3140 2300 840
609 3145 2302 843
average stairs needed steps minimum maximum
1 271.090 271.100 239 313
2 367.570 735.150 540 954
3 663.998 1992.003 1234 2911
4 1353.416 5413.674 2748 8980
5 2933.418 14667.099 6715 27210
6 6648.557 39891.354 18394 80050
7 15550.874 108856.130 42380 218892
calculated
1 170 270
2 317 734
3 633 1999
4 1333 5432
5 2933 14765
6 6673 40138
7 15573 109111
8 37063 296604
9 89573 806257
10 219154 2191640
Real time: 55.269 s
Phix
with javascript_semantics -- (nb expect blank screen for ~30s) integer total_seconds = 0, total_steps = 0 printf(1,"time behind ahead\n") printf(1,"---- ------ -----\n") for trial=1 to 10_000 do integer stairs_length = 100, steps_behind = 0, seconds = 0 while steps_behind < stairs_length do steps_behind += 1 for wizard=1 to 5 do steps_behind += (rnd()*stairs_length < steps_behind) stairs_length += 1 end for seconds += 1 if trial=1 and seconds>=600 and seconds<=609 then printf(1,"%4d %7d %6d\n", {seconds, steps_behind, stairs_length-steps_behind}) end if end while total_seconds += seconds total_steps += stairs_length if trial=1 then printf(1,"\n") end if end for printf(1,"Average seconds: %f, average steps: %f\n", {total_seconds/10000,total_steps/10000})
- Output:
time behind ahead ---- ------ ----- 600 1978 1122 601 1980 1125 602 1986 1124 603 1990 1125 604 1995 1125 605 1999 1126 606 2002 1128 607 2004 1131 608 2008 1132 609 2012 1133 Average seconds: 2913.609300, average steps: 14668.046500
Perl
<lang perl>#!/usr/bin/perl
use strict; # https://rosettacode.org/wiki/Long_stairs use warnings;
my $sumseconds = my $sumsizes = my $runs = 0; for ( 1 .. 1e4 )
{
$runs++;
my $behind = 0;
my $ahead = 100;
my $seconds = 0;
while( $ahead > 0 )
{
rand() <= ($ahead / ($behind + $ahead)) ? $ahead++ : $behind++ for 1 .. 5;
$behind++; # one step up
$ahead--;
$seconds++;
$_ == 1 and 600 <= $seconds <= 609 and
print "step $seconds: $behind behind, $ahead ahead\n";
}
$sumsizes += $behind;
$sumseconds += $seconds;
}
printf "\naverage stair length %d average seconds %d\n",
$sumsizes / $runs, $sumseconds / $runs;</lang>
- Output:
step 600: 2233 behind, 867 ahead step 601: 2237 behind, 868 ahead step 602: 2241 behind, 869 ahead step 603: 2247 behind, 868 ahead step 604: 2249 behind, 871 ahead step 605: 2254 behind, 871 ahead step 606: 2258 behind, 872 ahead step 607: 2262 behind, 873 ahead step 608: 2268 behind, 872 ahead step 609: 2273 behind, 872 ahead average stair length 15438 average seconds 3067
Python
<lang python>""" https://rosettacode.org/wiki/Long_stairs """
from numpy import mean from random import sample
def gen_long_stairs(start_step, start_length, climber_steps, add_steps):
secs, behind, total = 0, start_step, start_length
while True:
behind += climber_steps
behind += sum([behind > n for n in sample(range(total), add_steps)])
total += add_steps
secs += 1
yield (secs, behind, total)
ls = gen_long_stairs(1, 100, 1, 5)
print("Seconds Behind Ahead\n----------------------") while True:
secs, pos, len = next(ls)
if 600 <= secs < 610:
print(secs, " ", pos, " ", len - pos)
elif secs == 610:
break
print("\nTen thousand trials to top:") times, heights = [], [] for trial in range(10_000):
trialstairs = gen_long_stairs(1, 100, 1, 5)
while True:
sec, step, height = next(trialstairs)
if step >= height:
times.append(sec)
heights.append(height)
break
print("Mean time:", mean(times), "secs. Mean height:", mean(heights))
</lang>
- Output:
Seconds Behind Ahead ---------------------- 600 2122 978 601 2127 978 602 2130 980 603 2135 980 604 2139 981 605 2144 981 606 2149 981 607 2153 982 608 2157 983 609 2162 983 Ten thousand trials to top: Mean time: 2780.502 secs. Mean height: 14002.51
Raku
<lang perl6>my ($trials, $t-total, $s-total) = 10000;
say 'Seconds steps behind steps ahead';
race for ^$trials {
my $stairs = 100; my $location = 0; my $seconds = 0;
loop {
++$seconds;
++$location;
last if $location > $stairs;
for (1..$stairs).roll(5) {
++$location if $_ <= $location;
++$stairs;
}
say " $seconds $location {$stairs-$location}" if !$_ && (599 < $seconds < 610);
}
$t-total += $seconds; $s-total += $stairs;
}
say "Average seconds: {$t-total/$trials}, Average steps: {$s-total/$trials}";</lang>
- Sample output:
Seconds steps behind steps ahead 600 2143 957 601 2149 956 602 2153 957 603 2158 957 604 2164 956 605 2170 955 606 2175 955 607 2178 957 608 2183 957 609 2187 958 Average seconds: 2716.0197, Average steps: 13677.143
Wren
<lang ecmascript>import "random" for Random import "./fmt" for Fmt
var totalSecs = 0 var totalSteps = 0 var rand = Random.new() System.print("Seconds steps behind steps ahead") System.print("------- ------------ -----------") for (trial in 1..10000) {
var sbeh = 0
var slen = 100
var secs = 0
while (sbeh < slen) {
sbeh = sbeh + 1
for (wiz in 1..5) {
if (rand.int(slen) < sbeh) sbeh = sbeh + 1
slen = slen + 1
}
secs = secs + 1
if (trial == 1 && secs > 599 && secs < 610) {
Fmt.print("$d $d $d", secs, sbeh, slen - sbeh)
}
}
totalSecs = totalSecs + secs
totalSteps = totalSteps + slen
} Fmt.print("\nAverage secs taken: $h", totalSecs/10000) Fmt.print("Average final length of staircase: $h", totalSteps/10000)</lang>
- Output:
Sample run:
Seconds steps behind steps ahead ------- ------------ ----------- 600 2112 988 601 2115 990 602 2121 989 603 2126 989 604 2130 990 605 2135 990 606 2141 989 607 2146 989 608 2150 990 609 2155 990 Average secs taken: 2914.465 Average final length of staircase: 14672.325