5
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Lenny Fingers is out on the street again, hustling people with his shell games. This time, he's using words.

As you walk by, Lenny grabs you with an enthusiastic smile and shows you two words side-by-side, betting that he can switch their positions with just a few swipes. It seems impossible, but Lenny has long, crazy fingers and he's quick and he talks nonstop. Distracted by all the patter, you don't realize that he is able to hide multiple letters under his fingertips. This allows him to transfer 2, 3, or 4 letters at a time from one word to the other.

We took some covert video and played it back in slow motion. Watch carefully:

 TESTS            RAINING
   |            .   |
   |          .     |
   |      I,N,G     |
   |      .         |
   v    .           v
TESTINGS          RAIN
   |    .           |
   |      .         |
   |      T,E,T,G   |
   |          .     |
   v            .   v
 SINS             TREATING
   |            .   |
   |          .     |
   |      T,A,T,G   |
   |      .         |
   v    .           v
TASTINGS          REIN
   |    .           |
   |      .         |
   |      T,A,S     |
   |          .     |
   v            .   v
 STING            RETAINS
   |            .   |
   |          .     |
   |      R,E,I,S   |
   |      .         |
   v    .           v
RESISTING          TAN
   |    .           |
   |      .         |
   |      S,T,I,G   |
   |          .     |
   v            .   v
 RESIN            STATING
   |            .   |
   |          .     |
   |      T,A,T,G   |
   |      .         |
   v    .           v
RESTATING          SIN
   |    .           |
   |      .         |
   |       R,E      |
   |          .     |
   v            .   v
STATING           RESIN
   |    .           |
   |      .         |
   |      S,T,I,G   |
   |          .     |
   v            .   v
  TAN             RESISTING
   |            .   |
   |          .     |
   |      R,E,I,S   |
   |      .         |
   v    .           v
RETAINS           STING
   |    .           |
   |      .         |
   |      T,A,S     |
   |          .     |
   v            .   v
 REIN             TASTINGS
   |            .   |
   |          .     |
   |      T,A,T,G   |
   |      .         |
   v    .           v
TREATING          SINS
   |    .           |
   |      .         |
   |      T,E,T,G   |
   |          .     |
   v            .   v
 RAIN             TESTINGS
   |            .   |
   |          .     |
   |      I,N,G     |
   |      .         |
   v    .           v
RAINING           TESTS


Observations:
1. At each step, Lenny transfers 2, 3, or 4 letters from one word to the other. Each side remains a valid English word after extraction and insertion of the letters.
2. The letters can be extracted from anywhere in the word, and they can be inserted anywhere in the other word. They need not be contiguous.
3. The spacing of the letters does not need to be maintained since Lenny can easily adjust the spacing of his fingers. However, the ordering of letters must be maintained because Lenny can't change the ordering of his fingers.
4. The direction of movement does not need to alternate between right to left and left to right. At each step, Lenny can swipe in either direction. (See the eighth and ninth step above, for example.)


Now that you know how it's done, see if you can figure out his latest scam:

Lenny changes

    TRACED           REHEAT

into

    REHEAT           TRACED


BTW, as long as it doesn't involve money, Lenny is a pretty easy-going guy. He doesn't care about American and British spellings (THEATER or THEATRE) and allows words like HEARTED with old and new meanings.




Here is an even more lowbrow solution than the ones found by @GarethMcCaughan:

TRACED REHEAT
RETRACED HEAT
TRACE HEARTED
TERRACED HATE
RACED THEATER
RATCHETED EAR
RATED CHEATER
CRATERED HEAT
CATER HEARTED
CHARTERED EAT
HATER CREATED
CHEATER RATED
EAR RATCHETED
THEATER RACED
HATE TERRACED
HEARTED TRACE
HEAT RETRACED
REHEAT TRACED

$\endgroup$
  • $\begingroup$ Did you want the no-computers tag on this? $\endgroup$ – Gareth McCaughan Aug 19 at 22:26
  • $\begingroup$ @GarethMcCaughan – Please feel free to use computers. This is a tough one, so you might need all the help you can get! (Although I believe that the humans, if they use the hints I gave, might still have a slight edge over the computers.) $\endgroup$ – SlowMagic Aug 19 at 22:34
  • $\begingroup$ I find it extremely easy to miss possible moves when looking by eye. $\endgroup$ – Gareth McCaughan Aug 20 at 0:52
  • $\begingroup$ @GarethMcCaughan – This puzzle gives me a sinister delight for exactly that reason — it's just too unintuitive, even for a single step. I'm hoping there's someone out there who has unhuman intuition who can take out a pencil and paper and just solve it like a breeze! $\endgroup$ – SlowMagic Aug 20 at 1:02
3
$\begingroup$

Confession: the following is all found using computer help. (I did at least write the code to do it.) Further confession: earlier versions of this answer included a bunch of not-very-satisfactory solutions using very obscure words; you can find them in the edit history if you care.

Unsurprisingly, one can do this more quickly if one allows rarer words. Thus:

traced reheat -> retraced heat
retraced heat -> trace hearted
trace hearted -> tracheate red
tracheate red -> rache treated
rache treated -> re tracheated
re tracheated -> reheat traced

but I at least was not familiar with

RACHE, a kind of hunting dog.

More acceptable:

traced reheat -> retraced heat
retraced heat -> traced hereat
traced hereat -> terraced heat
terraced heat -> raced thereat
raced thereat -> ratcheted era
ratcheted era -> ached retreat
ached retreat -> reached treat
reached treat -> reach treated
reach treated -> re tracheated
re tracheated -> reheat traced

Simpler still but longer (and I regret that I've been too lazy to mark it up with exactly what letters are being shifted each way, this time):

traced reheat
retraced heat
traced hereat
terraced heat
raced thereat
ratcheted era
ached retreat
reached treat
reach treated
reteach rated
teach rerated
teacher rated
ear ratcheted
theater raced
hate terraced
hearted trace
heat retraced
reheat traced

$\endgroup$
  • $\begingroup$ Gareth, there must be something missing from your word list. We have confirmatory evidence of that. ;-) $\endgroup$ – SlowMagic Aug 20 at 0:56
  • $\begingroup$ There is a solution to this puzzle which uses only the most common words, if you can accept HEARTED. $\endgroup$ – SlowMagic Aug 20 at 0:57
  • $\begingroup$ @SlowMagic I think you may be commenting on an old version of my answer; the one here now uses only words that I think are no rarer than HEARTED. $\endgroup$ – Gareth McCaughan Aug 20 at 0:59
  • $\begingroup$ No, I am seeing your latest solution... And I will admit that there are at least three words which are making me ashamed of my public school education. $\endgroup$ – SlowMagic Aug 20 at 1:09
  • $\begingroup$ OK, then take a look at the latest addition which is a little lower-brow. $\endgroup$ – Gareth McCaughan Aug 20 at 1:24

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