Problem of the Month (September 2018)

Given a U.S. city name (without spaces), what is the largest number that can be represented by an equation, using some direct substitution of letters for digits or symbols? For example, NEWYORK, PHOENIX, and RALEIGH contain 7 different letters and therefore can represent 729=3^6. CHICAGO could represent 98=97+1. Only the 10 digits, the 4 arithmetic operations, an equal sign, a carat for exponentiation, and parentheses are allowed. The usual order of operations is used.

We can also ask the reverse question. Given n, what is the shortest U.S. city for which the maximum value on both sides of the equal sign is n?

Joe DeVincentis, George Sicherman, and Bryce Herdt sent solutions.

The best known solutions for the 100 most populated U.S. cities are shown below.

Largest Expressions for Cities
CityEquation
MESA
RENO
none
MIAMI98=98
OMAHA9=3×3
TAMPA(9)=9
TULSA
PLANO
BOISE
9=8+1
DALLASnone
AUSTIN
ELPASO
TUCSON
FRESNO
NEWARK
DURHAM
LAREDO
81=9^2
DENVER
BOSTON
8–9=–1
AURORA5^2=25
IRVINE
IRVING
10=1+9
TOLEDO48=6×8
NEWYORK
PHOENIX
RALEIGH
GILBERT
FREMONT
729=3^6
CHICAGO98=97+1
CityEquation
HOUSTON4^9=8^6 (JD)
SANJOSE2^7=128
SEATTLE79=88–9
DETROIT2×47=94 (JD)
MEMPHIS
ANAHEIM
4^4=256
ATLANTA98=9+89
OAKLAND
WICHITA
NORFOLK
GARLAND
128=2^7
LINCOLN8+79=87
BUFFALO
LUBBOCK
7+88=95
STLOUIS
ORLANDO
9^3=729
HIALEAH8+90=98
SANDIEGO
PORTLAND
CHANDLER
RICHMOND
8×95=760
COLUMBUS73×8=584
LASVEGAS
GLENDALE
675=9×75
HONOLULU96=6×4×4 (JD)
SANTAANAnone
STOCKTON36^4=6^8
CityEquation
CHARLOTTE
NASHVILLE
12×83=996 (JD)
FORTWORTH76×3=6×38
BALTIMORE
LONGBEACH
FORTWAYNE
8×972=6^5 (JD)
MILWAUKEE13×76=988 (JD)
ARLINGTON
LEXINGTON
3^8=9×729 (JD)
CLEVELAND4^8=8^5×2
RIVERSIDE81+98=179
SAINTPAUL4^93=8^62 (JD)
HENDERSON164×6=984
ANCHORAGE7569=87^2 (JD)
DESMOINES975=13×75 (JD)
LOSANGELES790×6=4740 (JD)
SANANTONIO1×9×9=89–8 (JD)
WASHINGTON2187×9=3^9 (JD)
LOUISVILLE2×614=1228 (GS)
SACRAMENTO972×7=6804 (JD)
KANSASCITY9×18×8=6^4 (JD)
NEWORLEANS8^63=2^189
PITTSBURGH7400=8×925 (JD)
CINCINNATInone (JD)
CityEquation
GREENSBORO4200=8×525 (JD)
JERSEYCITY6×81×2=972 (JD)
CHULAVISTA19683=27^3 (JD)
SCOTTSDALE31+993=4^5 (JD)
CHESAPEAKE815×7=5705 (JD)
BATONROUGE96×87=8352 (JD)
ALBUQUERQUE 0+1×9×8=9×8 (JD)
MINNEAPOLIS3^22×9=81^6 (JD)
BAKERSFIELD2×4^97=8^65
JACKSONVILLE9^6+7=531448 (JD)
SANFRANCISCO6^7=3^7×16×8 (GS)
INDIANAPOLIS1×617×7=4319 (GS)
OKLAHOMACITY6+390619=5^8 (GS)
STPETERSBURG 960–6–79=875 (GS)
WINSTONSALEM512^8=2^70×4 (JD)
VIRGINIABEACH43+531398=9^6 (GS)
CORPUSCHRISTI407×194=78958 (GS)
NORTHLASVEGAS(43×2)^9=86^9

Joe DeVincentis also wanted to do state capitals that had different letter patterns.

CityEquation
SALEM3^2=9 (GS)
HELENA64=4^3 (JD)
PIERREnone (JD)
AUGUSTA95=5×19 (JD)
CONCORD98=98+0 (JD)
LANSING1+97=98 (JD)
TRENTON97=89+8 (JD)
HARTFORD27^4=9^6 (JD)
FRANKFORT12^3=1728 (JD)
CityEquation
MONTGOMERY8^7=4^8×32 (JD)
LITTLEROCK7×887=6209 (JD)
CARSONCITY9^5=729×81 (JD)
CHARLESTON6^5=18×432 (JD)
MONTPELIER8740=92×95 (JD)
PROVIDENCE4^7=512×32 (JD)
HARRISBURG7544=8×943 (GS)
SPRINGFIELD512^3=8^9+0 (JD)
SALTLAKECITY8×6^6×1=72^3 (GS)
JEFFERSONCITY 43553×2=87106 (GS)

Shortest City for a Given Maximum Value
ValueEquationCity
–5565–561=–556 (JD)LILESVILLE, North Carolina
–2230–223×1=–223 (JD)MILLERSVILLE, Maryland
–15650–206=6–162 (JD)MONROE CENTER, Illinois
–1155–120=–115 (JD)FROSTPROOF, Florida
–112–8×14=6–118 (JD)ASHLEY FALLS, Massachusetts
–60–55–5=–60 (JD)ANNANDALE, Virginia
–1563/3–36=–15 (JD)POCONO PINES, Pennsylvania
–110–11=–11 (JD)PALL MALL, Tennessee
–10–10=10–20 (JD)BAR HARBOR, Maine
–57–12=–2–3 (JD)BETHLEHEM, Georgia
00=–0 ERIE, Pennsylvania
11=1×1 (JD)ORONO, Maine
22=450/225 (JD)LANEVILLE, Texas
47–3=11–7 (JD)PUYALLUP, Washington
55=1885/377 (JD)SCOTTS HILL, Tennessee
66=36/6 (JD)NEWNAN, Georgia
77=665/95 (JD)BECCARIA, Pennsylvania
88=2×2×2 (JD)GUALALA, California
1010=1+9IRVINE, California
112+9=11DUPREE, South Dakota
123+9=11+1 (JD)BOHANNON, Virginia
142×7=6006/429 (JD)ROCHELLE PARK, New Jersey
176+11=8+9 (JD)TOLLESON, Arizona
1818=108–90 (JD)INDIANOLA, Illinois
1919=19×1 (JD)AHSAHKA, Idaho
2264×77/224=22 (JD)CHAPPELL HILL, Texas
2424=288/12 (JD)ESTELLINE, South Dakota
ValueEquationCity
278+19=11+16 (JD)PINCONNING, Michigan
2828=112/4 (JD)LAKE ELMO, Minnesota
3636=6×6 (JD)ZAPATA, Texas
444×11=8+9×4 (JD)GREENSBURG, Indiana
46322/7=92/2 (JD)MOORESBORO, North Carolina
4747=88–41 (JD)WESTTOWN, New York
4848=6×8TOLEDO, Ohio
567×8=28×2 (JD)HOLUALOA, Hawaii
6464=4^3AUBURN, Alabama
6868=86–18 (JD)SARASOTA, Florida
728×9=9×8 (JD)GLENELG, Maryland
77–22+99=9+68 (JD)ABBOTTSTOWN, Pennsylvania
7979=88–9SEATTLE, Washington
8040+40=80 (JD)ECLECTIC, Alabama
8181=9^2AUSTIN, Texas
842×42=84 (JD)NIANGUA, Missouri
8686=78+8 (JD)OAKBORO, North Carolina
882×44=88 (JD)WINNETT, Montana
891+88=89 (JD)WHEELER, Illinois
9090=30×3 (JD)STANTON, Missouri
9191=1×91 (JD)TORONTO, Kansas
9292=46×2 (JD)MODESTO, California
9393=3×31 (BH)DUBUQUE, Iowa
942×47=94 (JD)DETROIT, Michigan
957+88=95BUFFALO, New York
968+88=96 (JD)MAMMOTH, Arizona
9797=89+8 (JD)TRENTON, New Jersey
9898=98MIAMI, Florida
999×11=99 (JD)SUCCESS, Arizona
10010^2=100 (JD)NEGAUNEE, Michigan

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 9/1/18.