Solution to the puzzle Mr. Game & Watch.

Mr. Game & Watch is one of the interesting puzzles of the Melbourne University Puzzle Hunt 2011 competition. According to the plot of the game that year the University was in danger, because 20 nefarious villains managed to infiltrate it, and the only defender of the University, who could stop them, mysteriously disappeared. However, you received his notebook with a description of the villains, as well as how you can stop their plans. But this is not so easy to do, because the owner of the notebook has encrypted its most valuable information using ingenious puzzles. Now you have to solve these puzzles to save the University.

Mr. Game & Watch is one of those villains. His description can be found in the fourth act of the game.

«Ever felt your life sometimes has too many coincidences? Ever felt that your life is almost like the Truman Show? If you have, it’s just possible that your life was temporarily taken over by Mr. Game & Watch. He will pick a target person at random, and then take perverse delight in subtly influencing the target’s life into a series of ridiculous coincidences. Often his targets are eventually led into a life of crime and debauchery. You can never be sure if your life has been toyed with by Mr. Game & Watch until he’s decided he’s had enough with you, and leaves his trademark symbol upon your bedside table. Just pray you never see it.
Last spotted in your mirror!»

Below you can find the corresponding puzzle with the same name. It was compiled by the main organizer of the Puzzle Hunt 2011, as well as the Vice-President of the Melbourne University Mathematics and Statistics Society (MUMS) at the time, Muhammad Adib Surani.

The puzzle consist of a large number of clocks of the same type: there are 24 of them in total, and they are neatly arranged on the page in six rows and four columns (we will number them from left to right and from top to bottom). In all  the clocks you can see a dial with 12 Roman numerals, the location of the hour hand and the minute hand, the time of day (AM or PM) and the angle between hands in degrees.

For all the clocks the hour can be easily determined directly from the position of the hands and the time of day. So it make sense to try to put these values together.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
hours 1 12 16 8 1 2 5 20 9 19 5 3 12 15 3 11 19 1 14 4 20 8 5 14

As you can see, the values are a variety of numbers from 1 to 20. At the same time the number of letters in the English alphabet is 26. Thus, each hour value can be associated with a single letter by simply numbering the alphabet. The result of such a simple manipulation in this case is a message.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
hours 1 12 16 8 1 2 5 20 9 19 5 3 12 15 3 11 19 1 14 4 20 8 5 14
letters A L P H A B E T I S E C L O C K S A N D T H E N

ALPHABETISE CLOCKS AND THEN…

However, the phrase is not completed. The next logical step in trying to complete it is to find amount of minutes for all the clocks. You can also try to determine it simply by the position of the corresponding hand, but there is a much more accurate method, since the angle between the hands is also given. Knowing this angle and the value of the hours it is possible to accurately calculate the value of the minutes.

Let's denote the hours as H, the minutes as M and the angle as ∠A. In addition, let's denote the angle from the start of the current hour to the hour hand as ∠H and the angle from the start of the current hour to the minute hand as ∠M.

The change in the position of the minute hand for each elapsed minute can be calculated as 360(degrees)/60(minutes) = 6(degrees per minute). Thus, ∠M = 6M.

The change in the position of the hour hand for each elapsed hour can be calculated as 360(degrees)/12(hours) = 30(degrees per hour). In addition to the number of hours H, that have already passed, it is also necessary to take into account the incomplete current hour: M(minutes)/60(minutes per hour) must be added to H.

Thus, ∠H = 30*(H + M/60) = 30H + (1/2)M.

The following property is true for all the clocks in the puzzle: the position of the minute hand can be obtained by adding the angle A to the angle H. Thus, we can set: ∠M* = ∠H + ∠A. Moreover, if ∠M* < 360°, then ∠M* = ∠M; and if ∠M* >= 360°, then ∠M* = ∠M + 360°. Thus, we have 2 formulas, in each of which we can express M.

1) ∠M = ∠H + ∠A; 6M = 30H + (1/2)M + ∠A; M = (2/11)*[30H + ∠A].

2) ∠M + 360 = ∠H + ∠A; 6M + 360 = 30H + (1/2)M + ∠A; M = (2/11)*[30H + ∠A – 360].

The second formula should be used for clocks numbered 8, 16 and 22. For all the other clocks the first formula should be used. Let's apply these formulas to the clocks and then round the result down to get the number of minutes.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
minutes 18 5 20 20 5 5 8 5 18 9 9 1 13 5 20 5 13 16 15 20 19 8 19 18

As you can see, all minute values also fall within the range from 1 to 26, and therefore to each of them can be assigned a corresponding letter in the same way.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
minutes 18 5 20 20 5 5 8 5 18 9 9 1 13 5 20 5 13 16 15 20 19 8 19 18
letters R E T T E E H E R I I A M E T E M P O T S H S R

However, this time they don't directly compose a message. Now it is necessary to use the fragment of the phrase obtained earlier: the clocks should be arranged in alphabetical order according to the letters corresponding to the values of the hours. If these letters coincide for several clocks, then they should be placed according to the order given in the task.

Original order:

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
hours A L P H A B E T I S E C L O C K S A N D T H E N
minutes R E T T E E H E R I I A M E T E M P O T S H S R

After sorting:

№ 1 5 18 6 12 15 20 7 11 23 4 22 9 16 2 13 19 24 14 3 10 17 8 21
hours A A A B C C D E E E H H I K L L N N O P S S T T
minutes R E P E A T T H I S T H R E E M O R E T I M E S

Now we have the end of the phrase: ...REPEAT THIS THREE MORE TIMES. Thus, the full hint is: ALPHABETISE CLOCKS AND THEN REPEAT THIS THREE MORE TIMES.

In this case the next logical step is to determine the number of seconds for all the clocks. This can be done by simply multiplying the fractional part of the minutes value by 60 and then rounding the result down.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
seconds 5 20 14 15 20 1 20 9 18 5 19 19 1 23 5 18 15 20 20 13 11 8 15 16

Then again it is necessary to assign a letter to each value.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
seconds 5 20 14 15 20 1 20 9 18 5 19 19 1 23 5 18 15 20 20 13 11 8 15 16
letters E T N O T A T I R E S S A W E R O T T M K H O P

And then - once again arrange the clocks in alphabetical order, this time in accordance with the values of the minutes. If these values coincide for some clocks, then they should be arranged according to the values of the hours, and if hours are also coincide, then according to the order given in the task.

Original order:

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
hours A L P H A B E T I S E C L O C K S A N D T H E N
minutes R E T T E E H E R I I A M E T E M P O T S H S R
seconds E T N O T A T I R E S S A W E R O T T M K H O P

After sorting:

№ 12 5 6 16 2 14 8 7 22 11 10 13 17 19 18 1 9 24 23 21 15 20 4 3
hours C A B K L O T E H E S L S N A A I N E T C D H P
minutes A E E E E E E H H I I M M O P R R R S S T T T T
seconds S T A R T W I T H S E A O T T E R P O K E M O N

We received another message: START WITH SEA OTTER POKEMON, which in itself make a little sense. However, in accordance with the hint received, this process of extracting a message should be repeated two more times.

Currently if we need time values less than a second for any purpose, we will probably not break second into 60 smoller pieces and so on, but divide it by some multiple of ten. However, history knows examples of the continuation of division by 60. Thus, in the Middle Ages the division of a hour into minutes, seconds, thirds and fourths was described, where each value consisted of sixty smaller values.

To solve the puzzle you could also try using the sexagesimal time values following the seconds. Their calculation is completely consistent with how we determined the number of seconds from the values for minutes: first, the fractional part of the previous time value should be multiplied by 60, after which the result must be rounded down.

Thus, thirds can be calculated from the seconds.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
thirds 11 13 7 12 1 20 25 20 4 12 5 18 1 4 20 20 9 18 1 5 8 15 18 1

Again all values fall within the range from 1 to 26, which make it possible to assign letters to them.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
thirds 11 13 7 12 1 20 25 20 4 12 5 18 1 4 20 20 9 18 1 5 8 15 18 1
letters K M G L A T Y T D L E R A D T T I R A E H O R A

Then come the next sorting phase, which give another message.

Original order:

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
hours A L P H A B E T I S E C L O C K S A N D T H E N
minutes R E T T E E H E R I I A M E T E M P O T S H S R
seconds E T N O T A T I R E S S A W E R O T T M K H O P
thirds K M G L A T Y T D L E R A D T T I R A E H O R A

After sorting:

№ 6 13 10 1 15 22 8 21 20 3 17 23 4 24 16 9 12 11 5 2 7 19 18 14
hours B L S A C H T T D P S E H N K I C E A L E N A O
minutes E M I R T H E S T T M S T R E R A I E E H O P E
seconds A A E E E H I K M N O O O P R R S S T T T T T W
thirds T A L K T O T H E G I R L A T D R E A M Y A R D

TALK TO THE GIRL AT DREAMYARD.

Next, in accordance with the hint, it remaind to repeat the process of extracting a message once more, this time for the fourths.

Based on the values of thirds, fourths can be calculated.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
fourths 19 20 20 9 15 9 5 15 22 21 9 12 5 5 20 14 1 5 12 23 8 1 2 22

Then they are matched with letters.

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
fourths 19 20 20 9 15 9 5 15 22 21 9 12 5 5 20 14 1 5 12 23 8 1 2 22
letters S T T I O S E O V U I L E E T N A E L W H A B V

After which the clocks are sorted again.

Original order:

№ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
hours A L P H A B E T I S E C L O C K S A N D T H E N
minutes R E T T E E H E R I I A M E T E M P O T S H S R
seconds E T N O T A T I R E S S A W E R O T T M K H O P
thirds K M G L A T Y T D L E R A D T T I R A E H O R A
fourths S T T I O S E O V U I L E E T N A E L W H A B V

After sorting:

№ 13 24 5 19 9 14 20 11 3 21 17 1 10 4 2 22 23 12 18 6 15 8 16 7
hours L N A N I O D E P T S A S H L H E C A B C T K E
minutes M R E O R E T I T S M R I T E H S A P E T E E H
seconds A P T T R W M S N K O E E O T H O S T A E I R T
thirds A A A A D D E E G H I K L L M O R R R T T T T Y
fourths E V O L V E W I T H A S U I T A B L E S T O N E

We received the last message: EVOLVE WITH A SUITABLE STONE.

It is worth noting that the described process of message extraction is easy to automate. The corresponding Python code can be found here.

Now let's take all three last messages together: START WITH SEA OTTER POKEMON, TALK TO THE GIRL AT DREAMYARD, EVOLVE WITH A SUITABLE STONE. Obtained hint pointed to the Pokemon Simisear from the Nintendo DS game Pokemon Black & White. The name of this Pokemon is the answer to the puzzle.

Shortly after the ending of the Puzzle Hunt 2011 an issue of Paradox magazine was published, which included an article by Muhammad Adib Surani about the game. In this article he summed up the results of the Puzzle Hunt this year and also talked about five of its puzzles, which the participants of the game liked most, according to their feedback. Among them was Mr. Game & Watch.

«This was one of the few puzzles that was created backwards: from an answer phrase into a puzzle. I picked SIMISEAR because STARMIE [another Pokemon] was a very nice counter [to it; this was used in the meta-puzzle], and then it just sort of merged itself with some clocks. I blame Professor Layton for this one, having just completed Professor Layton and the Unwound Future (a puzzle video game on the Nintendo DS) earlier this year. They had one too many puzzles which invovled clocks and angles, and I thought about how I could use it to give letters. It turns out that you can give five letters just by measuring the angle between the hour and minute hands to five decimal places.»

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