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, his notebook comes into your hands 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 could 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 was a 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 consisted of a large number of clocks of the same type: there were 24 of them in total, and they were neatly arranged on the page in six rows and four columns (we will number them from left to right and from top to bottom). All clocks featured 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 in degrees. The angle apparently represented the difference between the positions of the hands: if the angle was measured from the minute hand to the hour hand counterclockwise, then it was positive, and if the angle was measured clockwise, then it was negative.

For all clocks the hour could be easily determined directly from the position of the hands and the time of day. So it made 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 were 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 could be associated with a single letter by simply numbering the alphabet. The result of such a simple manipulation in this case was 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 was not completed. The next logical step in trying to complete it was to find amount of minutes for all the clocks. You could also try to determine it simply by the position of the corresponding hand, but there was a much more accurate method, since the angle between the hands was also given in the task. Knowing this angle and the value of the hours, it was possible to accurately calculate the value of the minutes.

How to do this? Let's start with a slightly different formulation of the problem. How, knowing the number of hours and minutes at a given time, calculate the angle between the hands? Let's denote hours by H, minutes by M, and angle by A. We will search angle A as the difference between the deviation of the minute hand from the start of the current hour and the deviation of the hour hand from the start of the current hour.

The change in the position of the minute hand for each passing minute can be calculated as 360(degrees)/60(minutes) = 6(degrees per minute). Thus, the deviation of the minute hand from the start of the current hour will be: 6 (degrees per minute) * M (minutes).

The change in the position of the hour hand for each passing 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, you must also take into account the current hour: you must add M(minutes)/60(minutes in an hour) to H. Thus, the deviation of the hour hand from the start of the current hour will be: 30(degrees per hour)*[H(hours) + M(minutes)/60(minutes per hour)].

Now all that remains is to calculate the angle as the difference between these two values: 

A(degrees) = 6(degrees per minute)*M(minutes) - 30(degrees per hour)*[H(hours) + M(minutes)/60(minutes per hour)]. 

Or without units of measurement: A = 6*M – 30*[H + M/60].

When using this formula, angle A will be positive if the minute hand is located further than the hour hand from the start of the current hour (if moving clockwise) and negative otherwise. This property is true for all clocks from the puzzle, except for clocks numbered 8, 16 and 22. For those clocks, according to our formula, we would get a large in absolute value negative angle, but the author of the puzzle preferred to use the corresponding positive angle instead. It can be easily calculated by simply adding 360 degrees to the negative angle. Accordingly, our formula in these cases undergoes a change:

A(degrees) = 6(degrees per minute)*M(minutes) - 30(degrees per hour)*[H(hours) + M(minutes)/60(minutes per hour )] + 360(degrees). 

Or without units of measurement: A = 6*M – 30*[H + M/60] + 360.

Now all that remains is to express the number of minutes M from both formulas: 

M = (2/11)*[30*H + A]; 

M = (2/11)*[30*H + A – 360].

Having this result, it was possible to accurately calculate the number of minutes for all clocks: to do so, one had to use the corresponding formula and then round the resulting value 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
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 the minute values also fell within the range from 1 to 26, and therefore to each of them could 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 did not directly compose the message. Now it was necessary to use the fragment of the phrase obtained earlier: the clocks had to be arranged in alphabetical order according to the letters corresponding to the values of the hours. If these letters coincided 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 sounded like: ALPHABETISE CLOCKS AND THEN REPEAT THIS THREE MORE TIMES.

In this case, the next logical step was to determine the number of seconds for all clocks. This could 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 was 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 it coincided for some clocks, then they should be arranged according to the values of the hours, and if it also coincided, 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 made a little sense. However, in accordance with the hint received, this process of extracting the message had to be repeated two more times.

Currently, if we need time values less than a second for any purpose, we will probably not break it into 60 smoller pieces and so on, but divide it by some multiple of ten. However, history also knows examples of the continuation of division by 60. Thus, in the Middle Ages, the division of an 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 could 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 fell within the range from 1 to 26, which made 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 came the next sorting phase, which gave 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 remained to repeat the process of extracting the message once more, this time for the fourths.

Based on the values of thirds, fourths were 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 were 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 were 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 was 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 also 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.»