# Problem of the Month (January 2017)

A Langford List is a list of numbers, each appearing twice, so that the number of the numbers in between the occurrences of n is n. The smallest such example is 3 1 2 1 3 2 (provided by Berend van der Zwaag). Define a Language Langford List to be a list of spellings of positive integers, each appearing twice, so that there are exactly n letters between the two occurrences of the number n. What is the smallest Langford List containing a given number? We can also do this in two dimensions. What is the smallest Language Langford Array containing a given number?

We can also use different counting systems. For each positive number, what is this shortest Roman Numeral Langford List, where the number of line segments between two numbers are counted? Or the shortest Digital Clock Langford List, where the number of LED segments between the two numbers are counted? Or the shortest Coinage Langford List, where the smallest number of US coins to represent that many cents are counted? By shortest, I mean the total number of letters, line segments, or coins.

Are there lists that contain each number 3 times, with the correct counts between neighboring pairs? What is the shortest example for each type?

Solutions were received from Mark Mammel, Bryce Herdt, and Alex Rower.

Alex Rowen found a Langford list with three of each number: 1, 9, 1, 6, 1, 8, 2, 5, 7, 2, 6, 9, 2, 5, 8, 4, 7, 6, 3, 5, 4, 9, 3, 8, 7, 4, 3

Here are the shortest known Language Langford lists containing a given number n:

3, 5, 7, 104, 8, 96, 1311, 16, 2312, 17, 2714, 2215, 2918, 21
THREE
TEN
THREE
SEVEN
TEN
FIVE
SEVEN
FIVE
FOUR
NINE
FOUR
EIGHT
NINE
FIVE
EIGHT
FIVE
SIX
ELEVEN
SIX
FOURTEEN
ELEVEN
THIRTEEN
FOURTEEN
EIGHT
THIRTEEN
EIGHT
ELEVEN
TWENTYTHREE
ELEVEN
SEVEN
SIXTEEN
SEVEN
TWENTYTHREE
SIXTEEN
SEVENTEEN
TWENTYSEVEN
TWELVE
SEVENTEEN
SIX
TWELVE
SIX
TWENTYSEVEN
NINE
TWENTYTWO
NINE
EIGHT
FOURTEEN
EIGHT
TWENTYTWO
FOURTEEN
SIX
TWENTY
SIX
TWENTYNINE
FIFTEEN
TWENTY
SEVENTEEN
FIFTEEN
TWENTYNINE
SEVENTEEN
NINE
TWENTYONE
NINE
EIGHTEEN
SEVENTEEN
TWENTYONE
EIGHTEEN
SEVENTEEN

19, 252024262830
NINETEEN
TWENTYTWO
TWENTYFIVE
NINETEEN
NINE
TWENTYTWO
NINE
TWENTYFIVE
TWENTY
TWENTYONE
TWENTYTHREE
TWENTY
NINE
TWENTYONE
NINE
TWENTYTHREE
NINE
TWENTYTWO
NINE
NINETEEN
TWENTYFOUR
TWENTYTWO
NINETEEN
FIFTEEN
TWENTYFOUR
SEVEN
FIFTEEN
SEVEN
TWELVE
TWENTYONE
SIX
TWELVE
SIX
TWENTYSIX
TWENTYONE
EIGHTEEN
SEVENTEEN
TWENTYSIX
EIGHTEEN
SEVENTEEN
SEVEN
SIXTEEN
SEVEN
TWENTYEIGHT
SIXTEEN
TWENTYFIVE
TWENTYSEVEN
TWENTYEIGHT
TEN
TWENTYFIVE
TEN
TWENTYSEVEN
EIGHT
EIGHTEEN
EIGHT
THIRTY
SIXTEEN
EIGHTEEN
THIRTEEN
SIXTEEN
THIRTY
THIRTEEN

Bryce Herdt found this 2-dimensional Langford Array using the consecutive numbers from 3 to 12:
 SIX ELEVEN THREE TWELVE ELEVEN SEVEN TEN FOUR SEVEN SIX FIVE THREE FIVE NINE EIGHT FOUR NINE EIGHT TEN TWELVE

Here are the shortest known Roman Numeral Langford Lists containing a given number n:

2, 5, 6 3, 9 4, 10 7 8 11 12, 15 13 14 16 17 18 19 20 21, 23 22, 25 24
V
VI
II
V
II
VI
VI
IX
III
VI
III
IX
IV
V
X
IV
V
VI
II
X
II
VI
(MM)
II
X
II
XV
VII
X
VIII
VII
XV
VIII
(MM)
VIII
XII
XV
VIII
IV
XII
IV
XV
(MM)
IX
XI
II
V
II
IX
V
XI
XII
XIV
XV
IV
XII
IV
XIV
XV
V
XIII
V
XII
XV
IX
XIII
XII
IX
XV
(MM)
XIV
V
VI
II
V
II
VI
XIV
XV
VII
XVI
X
VII
XV
II
X
II
XVI
(MM)
XI
XVII
XXI
XI
III
IX
III
XVII
IX
XXI
(MM)
XVIII
VI
IX
III
VI
III
IX
XVIII
VIII
XIX
XI
VIII
II
V
II
XI
V
XIX
(MM)
XX
IX
XI
II
V
II
IX
V
XI
XX
XXIII
XXXI
XXI
II
V
II
IX
V
XXIII
IX
XXI
XXXI
(BH)
? XXIV
XIV
V
VI
II
V
II
VI
XIV
XXIV
(BH)

Here are the shortest known Digital Clock Langford Lists containing a given number n:

4, 11, 125, 15, 286, 167, 8, 17, 259101314, 2618, 2919, 20, 2721, 22, 232430

(MM)

(MM)

(BH)

(MM)

(MM)

(MM)

(BH)

?

?

Bryce Herdt's solution for 6 and 16 contains only even numbers, and Mark Mammel's list for 13 only contains odd numbers. Bryce wondered whether there were lists having greatest common factor higher than 2.

Here are the shortest known Coinage Langford Lists containing a given number n:

1, 42, 3, 5, 678910

(MM)

(MM)

(MM)

(MM)

111213141516

(MM)

(MM)

(MM)

(MM)

(BH)

17181920

(MM)

(MM)

(MM)

(MM)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/1/17.