fusible numbers - mathpuzzle.com both ends of fuse a and one end of fuse b. ‣ fuse a burns out 30...
TRANSCRIPT
FusibleNumbers
JeffEricksonUIUCComputerScience
Anold(?)puzzle
‣ Youaregivenseveralfuses,eachofwhichburnforexactlyoneminute.
‣ Thefusesburndon’tburnuniformly,so
youcan’tpredicthowmuchfusewillbeleftafter(say)15seconds.
‣Howdoyoumeasureanintervalof45
seconds?
[AttributedtoCarlMorris]
!
"#$
%#&
Solution
Measuring30seconds:
Measuring45seconds:!
"#$‣ Lightbothendsofafusesimultaneously!
‣ Thefuseburnsout30secondslater.
‣ Weneedtwofuses.
‣ SimultaneouslylightbothendsoffuseAandoneendoffuseB.
‣ FuseAburnsout30secondslater;lighttheotherendoffuseB.
‣ FuseBburnsout15secondslater.
Whatelsecanwedowiththis?
Saythatarealnumberxisfusibleifonecanmeasurexminutesexactly,usingafinitenumberof1‐minutefuses.
‣ Fusescanbeliteitherattime0,orpreciselywhenanotherfuseburnsout.
‣ Anyfinitenumberoffuseendsmaybelitsimultaneouslyatthesetimes.
‣ Theintervalstartswhenfirstfuseislit,endswhenlastfusegoesout.
‣ Nocheating!Fusescan’tbecut,stopped,orlitinthemiddle;nootherclocks.
Fusiblenumbers
Ifwelightoneendattimeaandtheotherattimeb,where|a–b|<1,
thefuseburnsoutattimea~b:=(a+b+1)/2.(pronounced“afuseb”)
Moreformally...
Anumberxisfusibleifandonlyif‣ x=0or‣ x=a~bforsomefusiblenumbersaandbwith|a–b|<1
Smallexamples
0~0= 1/2
0~1/2= 3/4
0~3/4= 7/8
⋮
0~(1–2–n)= 1–2–(n+1)
0 1/2 3/4 7/8 15/16 31/32 …
Limitpointat1
a~b:=(a+b+1)/2
Smallexamples
1/2~1/2= 1
1/2~3/4= 9/8
1/2~7/8= 19/16
…
1/2~(1–2–n)= 5/4–2–(n+1)
1 9/8 19/16 39/32 …
Limitat5/4
a~b:=(a+b+1)/2
Smallexamples
1/2~1/2= 1
1/2~3/4= 9/8
1/2~7/8= 19/16
…
1/2~(1–2–n)= 5/4–2–(n+1)
1/2~1= 5/4
1/2~9/8= 21/16
…
1/2~(5/4–2–n)= 11/8–2–(n+1)
1 9/8 19/16 39/32 …
Limitat5/4
5/4 21/16 43/32 …
Limitat11/8
a~b:=(a+b+1)/2
Smallexamples
1/2~1/2= 1
1/2~3/4= 9/8
1/2~7/8= 19/16
…
1/2~(1–2–n)= 5/4–2–(n+1)
1/2~1= 5/4
1/2~9/8= 21/16
…
1/2~(5/4–2–n)= 11/8–2–(n+1)
1 9/8 19/16 39/32 …
Limitat5/4
5/4 21/16 43/32 …
Limitat11/8
1/2~(3/2–2–m–2–n)=3/2–2–m–2–(n+1)
Limitat23/16Limitat47/32
Doublelimitpointat3/2
11/8 45/32 …
23/16…
47/32 …
a~b:=(a+b+1)/2
Smallexamples
0 1/2 3/4 7/8 15/16 31/32 …
1 9/8 19/16 39/32 …
Limitat5/4
5/4 21/16 43/32 …
Limitat11/8Limitat23/16
Limitat47/32
Doublelimitpointat3/2
11/8 45/32 …
23/16…
47/32 …
Limitpointat1
Smallexamples
0 1/2 3/4 7/8 15/16 31/32 …
1 9/8 19/16 39/32 …
Limitat5/4
5/4 21/16 43/32 …
Limitat11/8Limitat23/16
Limitat47/32
Doublelimitpointat3/2
11/8 45/32 …
23/16…
47/32 …
Limitpointat1
Triplelimitat7/4
3/2 7/413/825/16 27/16
Smallexamples
0 1/2 3/4 7/8 15/16 31/32 …
1 9/8 19/16 39/32 …
Limitat5/4
5/4 21/16 43/32 …
Limitat11/8Limitat23/16
Limitat47/32
Doublelimitpointat3/2
11/8 45/32 …
23/16…
47/32 …
Limitpointat1
Triplelimitat7/4
3/2 7/413/825/16 27/16
Quadruplelimitat15/8
Smallexamples
0 1/2 3/4 7/8 15/16 31/32 …
1 9/8 19/16 39/32 …
Limitat5/4
5/4 21/16 43/32 …
Limitat11/8Limitat23/16
Limitat47/32
Doublelimitpointat3/2
11/8 45/32 …
23/16…
47/32 …
Limitpointat1
Triplelimitat7/4
3/2 7/413/825/16 27/16
Quadruplelimitat15/8
15/8 31/16 63/32 …
Quintuplelimitat31/16Sextuplelimitat63/32
Limitoflimitsoflimitsof...at2
Smallexamples
0 1/2 3/4 7/8 15/16 31/32 …
Limitpointat1
1 9/8 19/16 39/32 …
Limitat5/4
5/4 21/16 43/32 …
Limitat11/8Limitat23/16
Limitat47/32
Doublelimitpointat3/2
11/8 45/32 …
23/16…
47/32 …
Triplelimitat7/4
3/2 7/413/825/16 27/16
Quadruplelimitat15/8
15/8 31/16 63/32 …
Quintuplelimitat31/16Sextuplelimitat63/32
Limitoflimitsoflimitsof...at22
Ordinals
1 2 3 4 5 6 …
ω ω+1 ω+2 ω+3 …
2ω 2ω+1 2ω+2 …
3ω 3ω+1 …
4ω…
5ω …
ω2 ω3
2ω2ω2+1 3ω3
ω4 ω5 ω6 …
‣Ord(x+1)=ωOrd(x)foranyfusiblenumberx
‣Ord(n)=ωωnforanyintegern
‣ Thefusiblenumbersarewell‐ordered,withordertypeε₀=ωε₀
⋰ω}
Margin
‣m(x)=differencebetweenxandsmallestfusiblenumber>x
Ifx<0,thenm(x)=–x
Otherwise,m(x)=m(x–m(x–1))/2
‣ Recursivecallsgiveafuseconfigurationforsmallestfusible>x
Code!
Somesmallmargins
!
"#$
m(0)=2‐1
!
"
!#$
%#&
'#(
m(1)=2‐3 m(2)=2‐10
!
"#$
%#&'#(")#"*+#($"#"*&+#%$"!)#*&$$)#"$(&*)#$)*+*"#)"$$!&+#"!$&
Sowhat’sm(3)?
1,3,10,...?
‣ –log2m(0)=1
‣ –log2m(1)=3
‣ –log2m(2)=10
‣ –log2m(3)=
‣ –log2m(4)=
1,541,023,937
REALLYREALLYBIG!
(probablybetweenSkewes’#andGraham’s#)
Thanks,Martin!
2049/1024
961/5129/8
1/2 3/4
0 0 0 1/2
0 0
465/25615/16
3/4
0 1/2
0 0
7/8
0
0 225/128
3/4
0 1/2
0 0
7/8
0
3/4
0 1/2
0 0
7/8
0
105/64
49/323/4
0 1/2
0 0
21/163/4
0 1/2
0 0
1/2
0 0
9/8
1/2 3/4
0 0 0 1/2
0 0