© Mar Curcó Iranzo, MATH+
You can vote for the best challenge 2025 here: https://www.mathekalender.de/wp/best-challenge-2025/
Author: Mar Curcó Iranzo
Challenge
It’s the 24th of December, and to celebrate Christmas Eve the elves of Santa’s workshop are organizing a special poem competition. The rules of the competition are as follows.
Rule 1: If the poem has n stanzas, then each stanza must comprise n lines.
Rule 2: Each word ending a line in a stanza needs to be different from all other line-ending words within the stanza.
Rule 3: The lines of each stanza must end with the same words as the lines in the previous stanza, following the order given by the spiral permutation, which is described below.
For example, if we are writing a poem with 5 stanzas, the first stanza will have 5 lines, each of which will end in a different word. As in figure 1, we call the ending words by W_{1}, W_{2}, W_{3}, W_{4}, W_{5}, in order.
Figure 1: Schematic listing of the five ending words.
If we now draw a spiral starting from the last word W_{5} as indicated in figure 2, the order of the spiral meeting the words determines the order of appearance of the words in the next stanza.
Figure 2: The spiral permutation.
That is, stanza number two should have lines with ending words W_{5}, W_{1}, W_{4}, W_{2}, W_{3}, in this order. Next, the order of the ending words of stanza number three will be determined by drawing the spiral permutation with the end words of stanza number two. This iteration is represented in figure 3.
Figure 3: Iteration of the spiral permutation.
For a poem with $n$ stanzas -- and n lines, with n end words -- the spiral permutation Q_{n} can be described mathematically as follows: If in a certain stanza, which is not the last, a word ends line $m$, then in the next stanza it ends line
Q_{n}(m) =\begin{cases} 2m & \text { if } m \leq n/2 \\ 2n-2m+1 & \text { if } m > n/2.\end{cases}
Rule 4: The order of the ending words must be different in every stanza of the poem.
Notice that, with this last rule, we cannot write a poem with n stanzas that satisfies the rules of the competition for every number n. For example, for n=4, we would have the situation in figure 4.
Figure 4: Counterexample to rule 4 for n=4.
That is, after three iterations, we would end up with the same word order as in the first stanza. This would contradict the last rule of the competition.
If the poems of the competition have a maximum of 10 stanzas, what are all the possible numbers of stanzas that the poems can have?
Extra: We asked one of the elves to write a poem fulfilling all the rules of the competition as an example. See below:
Snow falls hard on the tall pine tree
Santa trips because he slips on milk
He asks the elves for help with the lost gift
Mrs. Claus calls them with a loud bell
And all eyes turn to Pip, the shy elf
Santa says, “Can you help us, Pip the elf?”
Pip nods and points at Santa’s map by the tree
The map shows numbers and arrows like a bell
Santa reads the map clue: “Find the cup of milk”
Rudolph finds it fast and wins a tiny gift
The milk clue contains a math puzzle gift
Can it be solved with the help of Pip the elf?
He sets down the riddle while Rudolph drinks the milk
The puzzle says, “Next clue under the tree”
Mrs. Claus laughs and rings her sweet bell
The clue reads “Footprints will bring you to the bell”
They are close to finding the lost gift!
Pip cheers and claps by the bright glowing tree
“Rudolph, stop licking milk!” shouts the shy elf
Santa looks around and sees footprints of milk
They rush outside behind the trace of milk
The prints lead them toward the chiming bell
Mrs. Claus claps, impressed with the shy elf
And with a soft smile, hands Pip the lost gift
Finally the gift is back, beside the glowing tree
Would you like to participate in the competition or share your own spiral poem?
- If you live in the Netherlands, you can send your poem to the author at m.curcoiranzo@gmail.com as part of the competition. You can submit your poem in english or dutch. Among all participants, the book Once Upon a Prime will be awarded three times as a prize.
- You can share your poem by posting it in the forum using the following link (in german or english): https://www.mathekalender.de/wp/forum/thread-1626.html
Among all correct entries, we will select a random participant who will also receive a small prize.
Possible Answers
- 2,3,5,6 and 7.
- 2,3,5,6 and 8.
- 2,3,5,6 and 9.
- 2,3,5,6 and 10.
- 2,3,5,7 and 9.
- 2,3,5,8 and 10.
- 2,5,6,7 and 8.
- 2,5,6,7 and 9.
- 2,5,6,7 and 10.
- 2,5,6,8 and 10.