Earlier today I set you this puzzle, created by Kirsty Land, a former student of King’s Maths School, the school that topped this year’s A-level rankings. (To read more about King’s Maths School, see my original post.)
The amazing word machine
The following rules transform one word to another
Add any vowel to the start of a word. i.e. MAZE –>AMAZE.
Add any consonant to the end of a word. i.e. CARD –> CARDS.
Delete two or more consecutive vowels. i.e. PLEASE –>PLSE.
Delete two or more consecutive consonants. i.e. STRING –> SING, RING, ING or STRI.
Double the entire word. i.e. AYE –> AYEAYE.
Note that any string of letters is considered a word. The “empty word” is a word with no letters, which we call “nothing”.
(a) Starting with LEAD, find a way to make GOLD
(b) Can all words disappear to nothing, that is, starting with any word, it is always possible to apply the above rules in such a way that you will get to the empty word? Prove the claim, or disprove it.
(c) Can you create any word from nothing, that is, starting with the empty word can you create any target word using only the rules above? Prove the claim, or disprove it.
Solution
a) Here’s one way. LEAD -> LD (rule 3) -> OLD (rule 1) -> OLDG (rule 2) -> OLDGOLDG (rule 5) -> OGOLDG (rule 4) -> OOGOLDG (rule 1) -> GOLDG (rule 3) -> GOLDGG (rule 2) -> GOLD (rule 4)
(b) Yes, all words can disappear to nothing. To start with, you can remove any consecutive vowels or consonants using rules 3 and 4. You can repeat this process until there are no consecutive vowels or consonants. We know this process must terminate because the word gets shorter at each step.
If we have reached the empty word, then we are done. Otherwise, the remaining word must alternate between vowels and consonants. Since the exact letters won’t matter for the next part of the answer, we will use the letter O to represent any vowel, and the letter X to represent any consonant. Having carried out the step above, you will be left with a word of one of the following forms:
OX…XO
XO…XO
XO…OX
OX…OX
In the first scenario, use rule 5 to double the word, then use rules 3 and 4 to delete the letters in pairs starting from the middle, until you get to the empty word.
In the second scenario, the use rule 1 to add a vowel to the front of the word. You are left with a string of the form we had in the previous scenario, which we can transform to the empty word as above.
The third scenario is much the same as the first. Use rule 5 followed by rules 3 and 4 to get to the empty word.
For the final scenario, the using rule 2 to add a consonant at the end, and then delete it using rule 4. Now we have a form like the one in the first scenario, which we know reduces to nothing.
c) Yes, you can create any word from nothing. Here’s one way. Start by using rules 1 and 2 repeatedly to create the following word:
AAEIOUUBBCDFGHJKLMNPQRSTVWXYZZ
We will call this sequence of letters one “block”. By using rules 3 and 4 to eliminate the unnecessary letters, it is possible to transform a block into any letter you like, or remove it entirely. For example, starting with such a block, you can get to the letter P as follows:
AAEIOUUBBCDFGHJKLMNPQRSTVWXYZZ
-> BBCDFGHJKLMNPQRSTVWXYZZ (rule 3 – note that it says you can remove two or more vowels. Here we removed seven at once)
-> PQRSTVWXYZZ (rule 4)
-> P (rule 4)
Note there are two copies of each of A, U, B, Z in the block, otherwise we wouldn’t be able to transform a block into an E, O, C or Y as we wouldn’t be able to remove the single vowel or consonant trapped on one side of our desired letter. Note also that this series of transformations can be applied to any block in a word, wherever it appears.
Before transforming blocks into letters, use rule 5 repeatedly to double the word, to get at least as many blocks as there are letters in the target word. For example, if you want to create SILVER from nothing, double three times to get a total of 8 blocks. Each block in turn is transformed into the desired letter in the target word, and finally all extraneous blocks are removed completely.
I hope you enjoyed today’s puzzle. I’ll be back in two weeks.
Thanks to King’s Maths School. During term time, the school posts a weekly maths challenge for curious 13-16-year olds, from which today’s puzzle was taken.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.
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