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)4Sr5
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