A small pointer oscillates (at ±85.5° right and left of the vertical center line) from which a small circular 'bullet' shoots off when you touch the screen. It gets reflected at the playing field's boundaries or any object in its way, moving at diminishing speed to reach the end of its trajectory where it grows to a circle as large as the boundary or other circles already present allow. Each new circle gets a value of 3 which diminishes by one count whenever it is hit by a bullet until it is shot down completely, giving you one point. The game is over when a bullet crosses, accidentally, a goal line at the bottom of the playing field a small distance above the pointer position.
So you collect points while new circles fill the playing field — three for each point you get if each bullet hits only one circle before it expires ... unless you aim the bullet cleverly to get multiple hits from each shot. Here is where the skill comes into the game. Since the pointer moves at constant angular speed to and fro you have to time the bullet release carefully to get the desired angle while estimating the bullet range at the same time. Try it.
There were 2594 participants at the time I collected the data in mid-March (3/18/2020 at 08:25) from the Ranking list accessible in the game app. One finds there the highest score each player listed has obtained. The graph below shows the number of players who have achieved a certain point score as a function of that score. Both the axes are logarithmically scaled with the actual number of players and points achieved given. For example, there were 103 players who achieved 8 points, and there were 374 who just got 1 point; there were also 387 people who didn't get any point but that cannot be plotted on a logarithmic scale. The more points you score the better you rank; the player with the most points having rank 1.
The graph shows that ranks 1 to 5 are held by 5 players, their point scores being 95, 69, 60, 58, and 55, respectively. The next lower score of 54 is shared by two players who rank #6 and #7 (the player who achieved that score earlier ranks higher), and so on.
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| Results for the game "321! – Angles Do Matter" by Antonio Jr. for the iOS devices iPhone and iPad. The plot shows the number of players who scored a certain number of points as a function of that number in a double logarithmic plot. (Data collected on 3/18/20 at 08:25) The lowest point score of 1 was achieved by 374 players; not shown are the 387 players who didn't get any point. The rates at which fewer players achieve higher scores appear to fall into five groups, A through E, which indicate increasingly higher difficulties as players get into higher point score regimes. The last several data points in each group fall on a straight line, that is, they follow (inverse) power laws with the slope of each being a measure of the group's difficulty. The slopes for the groups are as follows: A: –1.2, B: –2.1, C: –4.2, D: –5.2, E: –7.4 |
As expected, the higher the score the fewer players could achieve it, with the number of players going down disproportionately faster. What wasn't expected is that the number of players doesn't decrease smoothly over the whole range of achieved points; rather several regions on the graph can be discerned that have different decay characteristics. For instance, the smooth curve for the first five data (for points 1 ... 5, group A) gets interrupted by a nearly flat section of four scattered data points which is followed by a straight declining part (points 6 ... 13, group B). Group C (points 14 ... 21) emerges similarly, as does group D (with points 22 ... 31); and somewhat arbitrarily one might add group E (points 32 ... 42). Beyond that range, mainly single players, and rarely 2 or 3, have reached the same high score, as listed here:
rank 22 19 18 15 14 13 12 11 9 8 6 5 4 3 2 1
score reached 43 44 45 46 47 48 49 50 51 53 54 55 58 60 69 95
# of players 2 3 1 3 1 1 1 1 2 1 2 1 1 1 1 1
In this point group range, 43 ... 95, are 23 players. The number of players on record, as presented on the graph, was 2206, see the data given in the table below for the five groups. This number does not include those players who did not achieve even one point of which there were 387, bringing the total number of players listed in the rankings to 2593. (The data were collected on 3/18/20 at 08:25 h.)
All the groups except A start with some scatter in the number of players as points get higher but then that number falls off at a rate that is well approximated by the slope obtained from a least squares fit to the (double-logarithmic) data. These straight lines are also shown in the plot. The table gives the details for this analysis.
These observations are interpreted as follows. The playing field is filling up with circles as score points accrue and fewer players make additional points because the chances for the bullet to bounce back from a circle across the goal line. This dynamic change in the level of difficulty can be overcome by persistent practice in precisely aiming the bullet to touch several circles in sequence while also gaining skills in controlling the position and size of new circles on the playing field. It seems that once the next level is reached, the difficulty is initially somewhat smaller than at the end of the previous level, causing a slightly larger number of players populating the low point range of the new level. This observation is chosen as the criterion for determining the end of a level. Thus the ends of the regions can be considered breakpoints that lead to the next higher level; they are reached for the five groups A to E with score points 5, 13, 21, 31, and 42, respectively. An explanation for this feature may be that the entry into a higher level is accompanied by a reduction of the number of spheres crowding the playing filed. This is usually achieved by hitting several circles with a single bullet, reducing the value in each of several circles, or even gaining several points, with a single shot.
# of players 2 3 1 3 1 1 1 1 2 1 2 1 1 1 1 1
In this point group range, 43 ... 95, are 23 players. The number of players on record, as presented on the graph, was 2206, see the data given in the table below for the five groups. This number does not include those players who did not achieve even one point of which there were 387, bringing the total number of players listed in the rankings to 2593. (The data were collected on 3/18/20 at 08:25 h.)
All the groups except A start with some scatter in the number of players as points get higher but then that number falls off at a rate that is well approximated by the slope obtained from a least squares fit to the (double-logarithmic) data. These straight lines are also shown in the plot. The table gives the details for this analysis.
These observations are interpreted as follows. The playing field is filling up with circles as score points accrue and fewer players make additional points because the chances for the bullet to bounce back from a circle across the goal line. This dynamic change in the level of difficulty can be overcome by persistent practice in precisely aiming the bullet to touch several circles in sequence while also gaining skills in controlling the position and size of new circles on the playing field. It seems that once the next level is reached, the difficulty is initially somewhat smaller than at the end of the previous level, causing a slightly larger number of players populating the low point range of the new level. This observation is chosen as the criterion for determining the end of a level. Thus the ends of the regions can be considered breakpoints that lead to the next higher level; they are reached for the five groups A to E with score points 5, 13, 21, 31, and 42, respectively. An explanation for this feature may be that the entry into a higher level is accompanied by a reduction of the number of spheres crowding the playing filed. This is usually achieved by hitting several circles with a single bullet, reducing the value in each of several circles, or even gaining several points, with a single shot.
The fitted lines describe the level of difficulty of getting a higher score in each group: The slopes of these lines (or rather their integer value) can be taken as the group difficulty 1, 2, 4, 5, 7; they describe how fast the number of players decreases that achieve the next higher score. Extrapolating the fitted lines of each group beyond the group breakpoints down to the abscissa one obtains the score just one player would be able to achieve; these scores are listed in the last row of the table above. It is very interesting that this extrapolated value is essentially the same for the last three groups, namely ~45, as seen on the graph where the slopes of groups C, D, and E converge.
The interpretation of this last observation could be that, statistically, the difficulty of the game gets higher after each breakpoint, however, the skill level of the players has actually not improved. The number of players just drops off faster in each higher group because of the larger difficulty. The reason for this lack of improvement for levels D and E could be that it is just luck that gets one to the next higher level. However, in the lower level transitions, from A to B and B to C, such an improvement is observed, cf. the 1-player points row in the table.
From my (upern) own playing experience such luck consists, for example, in getting to a circle configuration in which more than one point can be scored by a single shot several times, clearing the playing field back to a configuration usually encountered only at much earlier stages. Another lucky streak could be initiated by a few circles ending up close to the pointer but without impeding the aiming and shooting so that it becomes less likely that a shot ends up ricocheting back and across the goal line. It is difficult to control such configurations on purpose.
Another observation that indicates that luck may play a role beyond skill is the thinning out of scores beyond 45 points: from 51 points up to 95 there are only 8 players that have a score, indicating that the "learning of skills" is not very smooth.
From my (upern) own playing experience such luck consists, for example, in getting to a circle configuration in which more than one point can be scored by a single shot several times, clearing the playing field back to a configuration usually encountered only at much earlier stages. Another lucky streak could be initiated by a few circles ending up close to the pointer but without impeding the aiming and shooting so that it becomes less likely that a shot ends up ricocheting back and across the goal line. It is difficult to control such configurations on purpose.
Another observation that indicates that luck may play a role beyond skill is the thinning out of scores beyond 45 points: from 51 points up to 95 there are only 8 players that have a score, indicating that the "learning of skills" is not very smooth.
Addendum
Incidentally, there was a major improvement to the scores obtained and some reshuffling of the ranks since the data for this blog were collected. On 5/7/20 a new point record was set with 108 points so that the previous top value of 95 points now ranks second. And on 5/20/20 the player who had held rank #5 with 55 points obtained a score of 65 points which places him/her now at rank #3.
For comparison with the situation of high-score players listed above (as of 3/18/20) here is the updated distribution of players and scores from 7/19/20 at ~10:00 h:
rank 22 21 18 17 16 14 13 11 10 8 7 5 4 3 2 1
score reached 44 45 46 47 48 49 50 51 53 54 58 60 65 69 95 108
# of players 2 1 3 1 1 2 1 2 1 2 1 2 1 1 1 1
# of players 2 1 3 1 1 2 1 2 1 2 1 2 1 1 1 1
Down to a score of 40 points the maximum number of players having the same score is 3.
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