Magazine article Word Ways

# Base Pairs

Magazine article Word Ways

# Base Pairs

## Article excerpt

Most people are familiar with the decimal system, in which ten different symbols (0 through 9) are used to represent numbers of any size. Our numbers are written in base 10, as distinct from base 2 which uses only the symbols 0 and 1. In base 2, the base 10 numbers 1 through 7 are rewritten as 1, 10, 11, 100, 101, 110, and 111. In base 16 (the hexadecimal system), used by computer programmers, the six letters A through F are added to the ten digits, resulting in "numbers" like AB9F which is equivalent to 43935 in base 10.

I consider a whole family of different bases, from base 11 (in which A is added) to base 36 (which contains the ten digits followed by A through Z). Some (but not all) of the "numbers" in these different bases will consist solely of letters, and even spell words. My object is to identify words in different bases which reduce to the same number in base 10. For example, one can show that GIRL in base 30 and BEER in base 34 both reduce to 449031 in base 10; these I call base pairs. Below I give a sampling of common base pairs of four or more letters. My investigation turned up relatively few pairs of five or more letters, and only one of six or more: GLOUTS and CASHES. There is one base triple: Lido 26 hale 28 akee 33.

The editor points out that base 26 words using letters only have been studied by Philip Cohen in the Feb 1977 Word Ways, and base 27 words (letters plus space) by Lee Sallows in May 1993.

```dice 24 alga 26  fife 24 cage 26  fold 25 bald 28    folk 25 balk 28
chap 28 back 29  mice 23 bare 29  gene 26 brag 29    keen 25 bran 30
fare 28 bale 31  farm 28 bait 31  dope 29 Bali 31    lode 25 Clio 30
lick 26 file 29  life 26 fins 29  harm 28 fish 29    flog 29 crop 31
ells 30 dale 31  gala 29 Enid 30  gogo 29 bead 33    chap 32 beck 33
dome 31 bees 33  dose 31 chit 32  core 32 blew 33    Leon 27 flak 30
etch 31 disk 32  pace 26 mice 27  palm 26 mild 27    girl 30 beer 34
plop 26 golf 30  fake 31 bled 34  blob 34 anew 35    duns 32 blow 34
cues 33 brow 34  Finn 31 cuts 33  flee 31 buds 34    hood 30 abed 36
jock 29 call 34  bite 35 Alps 36  mice 28 glib 31    gods 31 comb 34
dual 33 cons 34  fact 32 bows 35  Mona 28 fake 32    fete 32 cuds 34
inks 32 cute 34  hail 31 fogs 32  leap 29 high 31    cabs 35 barf 36
lick 29 deft 34  Lido 29 card 35  limn 29 cays 35    Celt 35 beta 36
joke 30 gale 32  hump 31 chit 35  kits 30 ills 31    Jane 31 hire 32
pail 29 knob 31  pone 29 dike 36  pops 29 dime 36    pork 29 dins 36
Nile 30 honk 33  nine 30 hope 33  nook 30 lids 31    gnaw 35 fact 35
flew 35 echo 36  miss 31 foul 35  roll 29 more 31    hard 34 fume 35
muss 31 hems 34  silo 29 gabs 35  jets 33 game 35    snip 29 jibe 33
lets 32 hugs 34  soon 29 hurt 34  mall 32 fool 36    malt 32 foot 36
mice 32 fury 36  moss 32 heal 35  peel 31 hobo 35    laps 33 hump 35
puke 31 glen 36  noel 32 gout 36  trap 30 kiln 34    roll 31 jays 35
sari 31 jogs 35  sash 31 John 35  sass 31 pore 32    nigh 33 lieu 34
noun 33 logs 34  taps 31 oath 33  tied 31 menu 34    Rome 32 palm 33
rots 32 past 33  roue 32 pate 33  rump 32 nail 34    pill 33 jowl 36
ruse 33 limb 36  rush 33 lime 36  sags 33 pubs 34    rows 34 pets 35
ride 35 pals 36  toad 35 racy 36  hosts 30 argue 34  spore 31 place 35

leaf 23 brie 28
fibs 29 cook 31
loop 26 both 32
gold 29 beef 33
limn 27 hens 29
plod 26 drip 32
felt 31 cram 33
hoot 30 abet 36
kelp 29 cone 34
curl 34 aver 36
deer 34 cape 35
John 30 gait 32
jeer 31 diet 35
jell 32 hogs 33
ceded 15 oats 30
pant 30 hear 34
nigh 31 legs 32
maul 32 fowl 36
salt 31 jobs 35
shag 31 purr 32
room 32 pans 33
tare 32 mesa 35