## Le Monde puzzle [#1072]

Posted in Books, Kids, R with tags , , , , , , , , , , , on October 31, 2018 by xi'an The penultimate Le Monde mathematical puzzle  competition problem is once again anti-climactic and not worth its points:

For the figure below [not the original one!], describing two (blue) half-circles intersecting with a square of side 20cm, and a (orange) quarter of a circle with radius 20cm, find the radii of both gold circles, respectively tangent to both half-circles and to the square and going through the three intersections. Although the problem was easily solvable by some basic geometry arguments, I decided to use them a minima and resort to a mostly brute-force exploration based on a discretisation of the 20×20 square, from which I first deduced the radius of the tangent circle by imposing (a) a centre O on the diagonal (b) a distance from O to one (half-)circle equal to the distance to the upper side of the square, leading to a radius of 5.36 (and a centre O at a distance 20.70 from the bottom left corner):

```diaz=sqrt(2)*20
for (diz in seq(5/8,7/8,le=1e4)*diaz){#position of O
```

In the case of the second circle I first deduced the intersections of the different circles by sheer comparison of black (blue!) pixels, namely A(4,8), B(8,4), and C(10,10), then looked at the position of the centre O’ of the circle such that the distances to A, B, and C were all equal:

```for (diz in seq(20*sqrt(2)-20,10*sqrt(2),le=1e4)){
distA=sqrt((diz/sqrt(2)-4)^2+(diz/sqrt(2)-8)^2)
```

even though Heron’s formula was enough to produce the radius. In both approaches, this radius is 3.54, with the position of the centre O’at 10.6 from the lower left corner. When plotting these results, which showed consistency at this level of precision,  I was reminded of the importance of plotting the points with the option “asp=1” in plot() as otherwise, draw.circle() does not plot the circles at the right location!

## a one-chance meeting [puzzle]

Posted in Books, Kids, pictures, R with tags , , , , , , on March 6, 2018 by xi'an This afternoon, I took a quick look at the current Riddler puzzle, which sums up as, given three points A, B, C, arbitrarily moving on a plane with a one-shot view of their respective locations, find a moving rule to bring the three together at the same point at the same time. And could not spot the difficulty.

The solution seems indeed obvious when expressed as above rather than in the tell-tale format of the puzzle. Since every triangle has a circumscribed circle, and all points on that circle are obviously at the same distance of the centre O, the three points have to aim at the centre O. Assuming they all move at the same velocity, they will reach O together… The question gets a wee bit more interesting when the number of points with the same one-time one-shot view option grows beyond 3, as these points will almost surely not all lie on a single circumscribed circle. While getting them together can be done by (a) finding the largest circle going through 3  points and containing all others [in case there is no such circle, adding an artificial point solves the issue!], triplet on which one can repeat the above instructions to reach O, and (b) bringing all points inside the circle to meet with one of the three points [the closest] on its straight-line way to O, by finding a point on that line at equal distance from both, a subsidiary question is whether or not this is the fastest way. Presumably not.  (Again I may be missing one item of the puzzle.)

When experimenting with a short R code, I quickly figured out that the circumscribed circles associated with all triplets do not necessarily contain all points. The resolution of this difficulty is however straightforward as it suffices to add an artificial point by considering all circumcentres and their distances to the farthest point, minimising over these distances and adding the extra point at random over the circumference. As in the example below. Incidentally, it took me much longer to write this post than to solve the puzzle, as I was trying to use the R function draw.circle, which supposedly turns a centre and a radius into the corresponding circle, but somehow misses its target by adapting the curve to the area being displayed. I am still uncertain of what the function means. I hence ended up writing a plain R function for this purpose:

```dracirc=function(A,B,C){
O=findcentr(A,B,C)
ro=dist(rbind(A,O))
lines(x=O+ro*sin(2*pi*seq(0,1,le=180)),
y=O+ro*cos(2*pi*seq(0,1,le=180)))}
```