Fun with numbers

I got a letter from John Gowland.  "Yes, I have made a Partridge puzzle. Yes, I am enjoying it. No, I have not yet solved it!  Seeing your starting point of Sr3 = (Sr)2, twenty years ago (my notes say 22 Jan ’80!) I thought it would be fun to express the sums of higher powers in terms of sums of lower powers. I can think of no use for them whatsoever, but I enjoyed doing them!"

 Sr4 = (1/5) Sr2(6Sr – 1) Sr5 = (1/3)[4( Sr)3 – (Sr)2] Sr6 = (12/7)( Sr)2Sr2 – (5/7) Sr4 Sr7 = 2(Sr)4 – Sr5 Sr8 = 8Sr2[Sr5 – (Sr)3] + Sr4 Sr9 = (4/5)( Sr)4[4Sr – 5] + (9/5) Sr5

Have you ever been curious about the mathematics behind the Eiffel Tower?

Rodolfo Kurchan of Puzzle Fun asked for the best packing of different square numbers into a square.  I managed to get the squares of (57, 59, 88, 89, 344, 444, 446, 614, 784, 806, 883, and 938) into a 6x6 square with only two black squares.  Can anyone do better?  Is it possible to make a perfect square square?

A nice result by Lew Baxter and Dean Hickerson:  f(n) = (2*10^(5n) - 10^(4n) + 2*10^(3n) + 10^(2n) + 10^n + 1)/3 provides a number whose cube lacks zeroes.  For example, f(7) = 66666663333334000000033333336666667.  When cubed, we obtain this number: 296296251851862962962481481592592598148148777777911111117777778814814892592598148148629629651851852962963

See f(99)^3 .

This week's puzzle is by Erich Friedman.  Using a word search format, fit the words ONE through EIGHT into a 5x5 square, so that the numbers read in any of the 8 cardinal directions. Solutions are below. Erich also mananged to put ONE through TWELVE into a 6x6 square.

How well can other languages be packed?  Matt Jones has pointed me to Numbers from 1 to 10 in Over 4000 Languages.
ENGLISH: one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve
SPANISH: uno, dos, tres, cuatro, cinco, seis, siete, ocho, nueve, diez
FRENCH: un, deux, trois, quatre, cinq, six, sept, huit, neuf, dix
GERMAN: eins, zwei, drei, vier, funf, sechs, sieben, acht, neun, zehn
ESPERANTO: unu, du, tri, kvar, kvin, ses, sep, ok, nau, dek

ENGLISH by Juha Saukkola, Daniel Scher, Sean Dawson, Roger Phillips, Mike Christie, Chris Lusby-Taylor, Erich Friedman
 E F S S T I O I E H G U X V R H R . E E T W O N E . F O U R F N X W . E I G H T S E V E N T H R E E R U O F .  F N X W .  E I G H T  S E V E N  T H R E E

A lot of languages by Roger Phillips

-------------------------------------------------------------
English

5x6 = ZERO ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE TEN

E F N N T
O I O I E O
W G U N V R
T H R E E E
T X I S Z

6x6 = ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE TEN ELEVEN TWELVE

T W E L V E
H S   T L E
G E E E E R
I V V N U H
E E I O W T
N N F X I S

7-triangles = ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE TEN

N        S I X   O N E
I E        E I G H T E
N S E        V F O U R
E E I R        E I W H
T V   X H        N V T
W E I G H T        I E
O N E F O U R        N

8-triangle = ONE TWO THREE FOUR FIVE SIX SEVEN EIGHT NINE TEN ELEVEN TWELVE

O
W N
T N E
W E N V
E V I F E
L E N G O S
V L   E H U I
E E H R T T R X

-------------------------------------------------------------
Esperanto

4x5 = UNU DU TRI QVAR QVIN SES SEP OK NAU

P E S E S
D U N U T
K V A R K
K V I N O

5x5 = UNU DU TRI QVAR QVIN SES SEP OK NAU DEK

K N     T
K V A R
P O I U
K E D N
S E S U

4x6 = UNU DU TRI QVAR QVIN SES SEP OK NAU DEK

N K O
N I P E S
K V A R D E
K U N U T S

3x7 = UNU DU TRI QVAR QVIN SES SEP OK NAU DEK

U N U T O P
K V A R D E K
K V I N S E S

6-triangle = UNU DU TRI QVAR QVIN SES SEP OK NAU

K
K O
U V N
P N A I
E D U R V
S E S T   K

-------------------------------------------------------------
French

5x6 = UN DEUX TROIS QUATRE CINQ SIX SEPT HUIT NEUF DIX

D E U X S X
Q N I C
T P E S O D
Q U A T R E
F H U I T

6x6 = UN DEUX TROIS QUATRE CINQ SIX SEPT HUIT NEUF DIX ONZE DOUZE

Q U A T R E
S N X I S Z
I T I U H N
O P D C E O
R E Z U O D
T S F

5x7 = UN DEUX TROIS QUATRE CINQ SIX SEPT HUIT NEUF DIX ONZE DOUZE

T O S I X S C
P N   F I D I
E Z U O D E N
S E R T A U Q
N T I U H X

7-triangle = UN DEUX TROIS QUATRE CINQ SIX SEPT HUIT NEUF

S
E I
P R O
T F T R
I S U A T
U I D E U X
H X C I N Q

8-triangle = UN DEUX TROIS QUATRE CINQ SIX SEPT HUIT NEUF DIX ONZE DOUZE

T R O I S T D S
Q N I C I I E
U D E U X P
A O H O T
T U N F
R Z U
E E
N

-------------------------------------------------------------
German

6x6 = EINS ZWEI DREI VIER FUNF SECHS SIEBEN ACHT NEUN ZEHN

S I E B E N
N E U N D A
I W C R C F
E Z E H N
I T U S
V   F

5x7 = EINS ZWEI DREI VIER FUNF SECHS SIEBEN ACHT NEUN ZEHN

S N I E
F Z E H N
U U W A C H T
N S I E B E N
F D R E I V S

8-triangle = EINS ZWEI DREI VIER FUNF SECHS SIEBEN ACHT NEUN ZEHN

E
I V
N E I
S F W E
H U Z Z R
C N U E N D
E F A C H T
S I E B E N

-------------------------------------------------------------
Spanish

5x6 = UNO DOS TRES CUATRO CINCO SEIS SIETE OCHO NUEVE

D O S I E S
C I N C O
S E R T H
O R T A U C
E V E U N O

6x6 = UNO DOS TRES CUATRO CINCO SEIS SIETE OCHO NUEVE DIEZ ONCE DOCE

S I E S   O
C I N C O H
E V E U N C
C U A T R O
O N D I E Z
D O S E R T

7-triangle = UNO DOS TRES CUATRO CINCO SEIS SIETE OCHO

S
E E
R T I
T E O S
C I N C O
S U   H D
C U A T R O

8-triangle = UNO DOS TRES CUATRO CINCO SEIS SIETE OCHO NUEVE DIEZ ONCE

C U A T R O
D Z C I N C O
O E O C H O
S I E S N
I D E U
E R E
T V
E

-------------------------------------------------------------
Italian

5x7 = UNO DUE TRE QUATTRO CINQUE SEI SETTE OTTO NOVE DIECI

E U Q N I C
U N O V E E
E T T E S
Q U A T T R O
D I E C I O E

7-triangle = UNO DUE TRE QUATTRO CINQUE SEI SETTE OTTO

Q E U Q N I C
U T D U E U
A T E S N
T E R O
T S T
R T
O

8-triangle = ZERO UNO DUE TRE QUATTRO CINQUE SEI SETTE OTTO NOVE DIECI

D U E U Q N I C
I O R E Z E O
E T T E S R
C T V   T
I O   T
N N A
U
Q

-------------------------------------------------------------
Dutch

5x5 = EEN TWEE DRIE VIER VIJF ZES ZEVEN ACHT

T H C A N
D T W E E
R E I V
F J I V E
S E Z

5x6 = EEN TWEE DRIE VIER VIJF ZES ZEVEN ACHT NEGEN TIEN ELF

T N E G E N
H N E V E Z
C E I I W E
A E R J T S
R D   F L E

6x6 = EEN TWEE DRIE VIER VIJF ZES ZEVEN ACHT NEGEN TIEN ELF TWAALF

T W A A L F
H D   R L J
C S R E T I
A E N I W V
Z E V E N
N E G E N

7-triangle = EEN TWEE DRIE VIER VIJF ZES SEVEN ACHT NEGEN

E E N E G E N
E F J I V E
W E S I V
T I E E
H R Z
C D
A

-------------------------------------------------------------
Portuguese

4x6 = UM DOIS TRES QUATRO CINCO SEIS SETE

S O C N I C
I E S I E S
O R T A U Q
D T R E S M

5x6 = UM DOIS TRES QUATRO CINCO SEIS SETE OITO NOVE DEZ

Q U A T R O
Z M D   C T
E V O N R I
D S I E S O
C S E T E

6x6 = UM DOIS TRES QUATRO CINCO SEIS SETE OITO NOVE DEZ ONZE DOZE TREZE

E V O N Z E
O C N I C S
E Z E R T I
Q U A T R O
M D O Z E D
S I E S S

6-triangle = UM DOIS TRES QUATRO CINCO

O R T A U Q
C S R M
N I E
I O S
C D

7-triangle = UM DOIS TRES QUATRO CINCO SEIS SETE OITO NOVE

D T O C N I C
O R T A U Q
I E I N M
S S O E
I V T
E E
S

8-triangle = UM DOIS TRES QUATRO CINCO SEIS SETE OITO NOVE DEZ ONZE DOZE

Q U A T R O D T
Z M     C O R
E V O N Z E
D S I E S
C T I
E O
S D
Maori

5x5 = TAHI RUA TORU WHAA RIMA ONO WHITU WARU IWA

T O R U
N I U T
O M R A I
A A H W H
W I   I W

5x6 = TAHI RUA TORU WHAA RIMA ONO WHITU WARU IWA TEKAU

W T A H I A     T A H I   U
H H O N O M     E U R O T
I W A R U I     K R   I N
T E K A U R     A A H W M O
U               U W   A   A

6-triangle = TAHI RUA TORU WHAA RIMA ONO WHITU

T R T O N O
O I A W W
R M H H
U A I
A T
U

7-triangle = TAHI RUA TORU WHAA RIMA ONO WHITU WARU IWA TEKAU

I R U O N O
H U R I M A
A A A H W
T K W H
O E I
R T
U

-------------------------------------------------------------
Danish

4x4 = EN TO TRE FIRE FEM SEKS SYV

V Y S E
O M K R
T R E I
N S F

4x5 = EN TO TRE FIRE FEM SEKS SYV OTTE NI TI

V Y S E
T E R T
K I N T
S F E M O

6-triangle = EN TO TRE FIRE FEM SEKS SYV OTTE NI TI

V Y S E K S
E R I F
T N E
T M
I O

-------------------------------------------------------------
Swahili

5x5 = MOJA MBILI TATU NNE TANO SITA SABA NANE

S N T S
I T A T U
T B N N E
A J O M E
M B I L I

5x6 = MOJA MBILI TATU NNE TANO SITA SABA NANE TISA KUMI

M N U T A T
A B A S A
J I N N E
T O L E
A K U M I

6-triangle = MOJA MBILI TATU NNE TANO SITA

S M B I L I
I O N O
T J N
A A E
T
U

7-triangle = MOJA MBILI TATU NNE TANO SITA SABA NANE TISA KUMI

A N N E N A N
S M B I L I
I A J O M
T B N U
A A K
T S
U

-------------------------------------------------------------
Mandarin

3x4 = YI ER SAN SI WU LIU QI BA

B U S Q
W A I Y
N R E L

4x4 = YI ER SAN SI WU LIU QI BA JIU SHI

J I U Q
S H I W
R A L Y
B E N

3x5 = YI ER SAN SI WU LIU QI BA JIU SHI

Q U I L R
W I H B E
Y J S A N

5-triangle = YI ER SAN SI WU LIU QI BA JIU

W Q S A N
U I L B
I Y R
J E

6-triangle = YI ER SAN SI WU LIU QI BA JIU SHI

E J I U W
R S H I Q
B A Y L
N

-------------------------------------------------------------
Nepali

4x5 = EK DUI TIN CHAR PANCH CHA SET AT NAU DAS

K S R I
E N A U
T I N H D
P A N C H

6-triangle = EK DUI TIN CHAR PANCH CHA SET AT NAU DAS

N I T E S C
P A N C H
D A S A
U K R
I E

-------------------------------------------------------------
Haitian Creole

4x4 = YOUN DE TWA KAT SENK SIS

A W T S
A D I
K N E S
Y O U N

4x5 = YOUN DE TWA KAT SENK SIS SET UIT NEF DIS

U I T S S
T W E I F
A N S D E
K Y O U N

5-triangle = YOUN DE TWA KAT SENK SIS

A Y O U N
W S I S
T D E
A N
K

6-triangle = YOUN DE TWA KAT SENK SIS SET UIT NEF DIS

Y O U N E F
I T S D
T W E I
A N S
K I
S

-------------------------------------------------------------
Esperanto (with K instead of Q)

4x5 = UNU DU TRI KVAR KVIN SES SEP OK NAU

P E S E S
O U N U T
K V A R D
N I V K

5x5 = UNU DU TRI KVAR KVIN SES SEP OK NAU DEK

K N     T  (same as Esperanto with Qs)
K V A R
P O I U
K E D N
S E S U

4x6 = UNU DU TRI KVAR KVIN SES SEP OK NAU DEK

P D S
E E U N U T
S K V A R O
N I V K

6-triangle = UNU DU TRI KVAR KVIN SES SEP OK NAU DEK

O K N I R T
K V A R S
E I U E
D N S
U E
P

-------------------------------------------------------------
Yucatec (classical)

3x5 = HUN CA OX CAN HO UAC UUC UAXAC

C   C
U A X A C
U O H U N

4x5 = HUN CA OX CAN HO UAC UUC UAXAC BOLON LAHUN

U U C A U
L A H U N
N O L O B
U A X A C

5-triangle = HUN CA OX CAN HO UAC UUC UAXAC

U A X A C
U C O H
C A U
N

6-triangle = HUN CA OX CAN HO UAC UUC UAXAC BOLON LAHUN

N N O L O B
U A X A C
H C A U
A O U
L H

-------------------------------------------------------------
Volapük

4x4 = BAL TEL KIL FOL LUL MAL VEL

L B M K
V U A I
T E L L
F O L

4x5 = BAL TEL KIL FOL LUL MAL VEL JOL ZUL DEG

J G E D B
F O L A M
Z U L E V
L I K   T

6-triangle = BAL TEL KIL FOL LUL MAL VEL JOL ZUL

B A L I K
M A L E T
Z U L V
L O F
J

7-triangle = BAL TEL KIL FOL LUL MAL VEL JOL ZUL DEG

B A L
F O L Z K
J A U I
M L L
D E G
V T

-------------------------------------------------------------
Klingon

4x6 = wa cha wej loS vagh jav Soch chorgh Hut

w v v S
e a h c o S
j g t u H l
c h o r g h

5x6 = wa cha wej loS vagh jav Soch chorgh Hut wamaH

l o S   h
c h o r g h
t c v a j
u h   v e
H a m a w

6-triangle = wa cha wej loS vagh jav Soch chorgh

c h o r g h
a w e j
h g a v
c v l
o o
S

7-triangle = wa cha wej loS vagh jav Soch chorgh Hut wamaH

l o S o c h a
c h o r g h
H a m a w
u   v e
t   j
a
v

-------------------------------------------------------------
Hawaiian

5x5 = EKAHI ELUA EKOLU EHA ELIMA EONO EHIKU

E H I K U
A M I L E
O N O E
E K A H I
E L U A

5x6 = EKAHI ELUA EKOLU EHA ELIMA EONO EHIKU EWALU EIWA UMI

E E L U A A
K O H A M W
A   N I H I
H U L O K E
I E W A L U

7-triangle = EKAHI ELUA EKOLU EHA ELIMA EONO EHIKU

U L O K E
E K A H I
E L I M A
O U H H
N A E
O

8-triangle = EKAHI ELUA EKOLU EHA ELIMA EONO EHIKU EWALU EIWA UMI

O N O E K A H I
E H I K U
H W I O U
A M I L E
U   A U
L W
E

Tok Pisin

5x5 = WAN TU TRI FOA FAIV SIKIS SEVEN ET NAIN TEN

A U T E N
O W R A E
F A I V V
N     E
S I K I S

4x6 = WAN TU TRI FOA FAIV SIKIS SEVEN ET NAIN TEN

W N E V E S
A S I K I S
N A R A O F
F U T E N

6-triangle = WAN TU TRI FOA FAIV SIKIS SEVEN ET

V S I K I S
I   U R E
A W T V
F A E
O N
A

7-triangle = WAN TU TRI FOA FAIV SIKIS SEVEN ET NAIN TEN

U S I K I S
T E N     E
R F A I V
I O I E
W A N

-------------------------------------------------------------
Balinese

5x6 = SA DUA TELU EMPAT LIMA ENEM PITU AKUTUS ASIA ADASA

A S A D A M
E M P A T E
D   I   E N
U S T L L E
A K U T U S

7-triangle = SA DUA TELU EMPAT LIMA ENEM PITU AKUTUS ASIA

T E L U D U A
A K U T U S
P A S I A
M M   P
E I
N L
E

8-triangle = SA DUA TELU EMPAT LIMA ENEM PITU AKUTUS ASIA ADASA

A K U T U S
S   T L T
D U A S I A
L D M P
E A M
T E
N
E

-------------------------------------------------------------
Javanese

5x6 = SIJI LORO TELU PAPAT LIMA ENEM PITU WOLU SANGA

U M I J I S
U L E T A O
T I O N   R
I M G W E O
P A P A T L

5x7 = SIJI LORO TELU PAPAT LIMA ENEM PITU WOLU SANGA SEPULUH

S E P U L U H
I A T E L U O
J   N O I T R
I E W G M I O
M T A P A P L

7-triangle = SIJI LORO TELU PAPAT LIMA ENEM PITU WOLU

O   P A P A T
R S M E N E
O I   U L
L J T U
I L
P O
W

8-triangle = SIJI LORO TELU PAPAT LIMA ENEM PITU WOLU SANGA SEPULUH

S E P U L U H W
S T L O R O
L I M A S L
P J E A U
A I N L
P G E
A T
T

-------------------------------------------------------------
Malagasy

5x6 = IRAY ROA TELO EFATRA DIMY ENINA FITO VALO SIVY

A D I M Y O
N O L A V L
I T R   I E
N I     S T
E F A T R A

6x6 = IRAY ROA TELO EFATRA DIMY ENINA FITO VALO SIVY FOLO

Y V I S
E F A T R A
F L N F A O
O D I M Y R
L T N
O L E T

5x7 = IRAY ROA TELO EFATRA DIMY ENINA FITO VALO SIVY FOLO

O L O F
L E N I N A S
E F A T R A I
T     O L A V
A D I M Y

7-triangle = IRAY ROA TELO EFATRA DIMY ENINA FITO VALO

D I M Y O   V
E F A T R A
N R I   L
I F R O
N O L
A E
T

8-triangle = IRAY ROA TELO EFATRA DIMY ENINA FITO VALO SIVY FOLO

E F A T R A
N O T I F   S
I L E R   I
N O L A V
A   O Y
R M
I
D

-------------------------------------------------------------
Tagalog

4x7 = ISA DALAWA TATLO APAT LIMA ANIM PITO WALO SIYAM

D A L A W A W
I N P I T O A
S I Y A M I L
A M T A T L O

6x6 = ISA DALAWA TATLO APAT LIMA ANIM PITO WALO SIYAM SAMPU

D A L A W A
U P M A S
A I I L I
T A T L O Y
O     A
A N I M

5x7 = ISA DALAWA TATLO APAT LIMA ANIM PITO WALO SIYAM SAMPU

D A L A W A S
U P M A S I
A I I L   Y
T A T L O   A
O A N I M

4x8 = ISA DALAWA TATLO APAT LIMA ANIM PITO WALO SIYAM SAMPU

M   T A T L O
S I Y A M I L T
N U P M A S I
D A L A W A   P

7-triangle = ISA DALAWA TATLO APAT LIMA ANIM PITO WALO SIYAM

M O T I P
D A L A W A
Y M P T
M I N A
L S T
L
O

8-triangle = ISA DALAWA TATLO APAT LIMA ANIM PITO WALO SIYAM SAMPU

D A L A W A L O
N   O U L
I   T P T
M L I M A
A P A T
Y S
I
S

-------------------------------------------------------------
Korean

4x6 = HANA TUL SET NET TASOT YOSOT ILGOP

A P O G L I
N T O S O Y
A L U T E N
H T A S O T

5x5 = HANA TUL SET NET TASOT YOSOT ILGOP

A N A H Y
I L G O P
L   S
U O T E N
T A S O T

5x6 = HANA TUL SET NET TASOT YOSOT ILGOP YODOL AHOP YOL

P Y O S O T
P O H A N A
D G T E S
O U L T O
L O Y I T

7-triangle = HANA TUL SET NET TASOT YOSOT ILGOP YODOL AHOP

T A S O T U L
L N N E T I
O A S O L
D H S G
O O O
Y P

8-triangle = HANA TUL SET NET TASOT YOSOT ILGOP YODOL AHOP YOL

P O G L I
A T O S O Y
N E T Y
T A S O T
H D U
O L
L P

-------------------------------------------------------------
Tibetan

4x5 = CHIG NYEE SUM ZHI NGA DUG DHUN

E   Z D C
E D H H M
Y U I   U
N G A   S

5x5 = CHIG NYEE SUM ZHI NGA DUG DHUN GYEY GU CHU

Y E Y G
Z M U S
N U H D A
C H I G
E E Y N

4x6 = CHIG NYEE SUM ZHI NGA DUG DHUN GYEY GU CHU

D Y Z E C
D H M E H H
U Y U Y I
N G A S G

6-triangle = CHIG NYEE SUM ZHI NGA DUG DHUN

M U S A
C H I G
D H U N
Z D Y
E
E

7-triangle = CHIG NYEE SUM ZHI NGA DUG DHUN GYEY GU CHU

N     Z
M U S H A
Y C H I G
E H D N
Y U Y
G E
E

-------------------------------------------------------------
Cantonese

4x5 = YAT YIH SAAM SEI NGH LUHK CHAT

S K C
E A H G N
I I A U
Y A T M L

5x5 = YAT YIH SAAM SEI NGH LUHK CHAT BAAT GAU SAHP

B K T C S
S A H P E
Y A A U I
T I A T L
N G H M

4x6 = YAT YIH SAAM SEI NGH LUHK CHAT BAAT GAU SAHP

L N S T T S
U A G A A E
H I Y H A I
K   P C B M

6-triangle = YAT YIH SAAM SEI NGH LUHK CHAT

T A H C
S A A M
L E Y H
U I G
H N
K

7-triangle = YAT YIH SAAM SEI NGH LUHK CHAT BAAT GAU SAHP

P C H G N
K H U L B
S A A M A
T G S A
H E T
I A
Y