Monthly Archives: April 2012

the wrong request – intereferences

so, what if instead of being asked to implement solutions, you were instead asked to solve problems?

a bad boss is that who thinks (s)he knows or is capable enough to take certain types of decisions by his/her own, hiding the actual problem from his employees, who then have to deal with a solution of his/her choice.

thing is, more often than not bosses are not there for thinking. in fact, bosses probably became bosses exactly cause they cannot solve these sort of problems (anymore). they do solve other types of problems, but not these. and the fact that they might believe they can, is often a problem (or leads to a non optimal solution).

so, a good boss is that who exposes the raw problem to his people and trusts their judgement and creative problem solving skills. that boss might even believe in people’s self-management. i think the formula is very simple – trust on your people’s experience, skills and intelligence and delegate on them. you as a boss take the other decisions, which you probably enjoy a lot doing, and on what you are in fact better than anyone else (that’s why you the boss). then you have good ingredients for success.

it’s always a good exercise to remember people to tell you in raw words what actually needs to be solved in the first place, not just what they thought it had to be done to solve it.


… i am one of the most systematically happy thing on earth!

and i get this realization about every other day.

maths – that endless game

…or may be not?

the fact that in the previous game we got 4 different graphs is of course a consequence of using a decimal positional system to write our numbers. what if we were using an hexadecimal system? or if we decided to work in base 13 instead? how many distinct graphs would there be for each base? how long would the transient states be before the graphs start to repeat, if they were actually guaranteed to repeat?

damn, see?, there’s always new questions to answer. that’s the (beautiful) problem with playing around with maths – once you start, you can’t stop, cause there is always a new game to be played as soon as you finish with the current one.

of course i couldn’t resist answering these questions. so, here we go, i computed all the graphs for all bases from base 2 (binary) to base 31. and indeed, for a given base, the sequence of graphs seems to get periodic. these are the results:

base: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
# graphs: 1 2 3 4 2 6 4 7 4 10 3 12 6 4 7 16 7 18 5 6 10 22 4 21 12 20 7 28 4 30
period: 1 2 2 4 2 6 2 6 4 10 2 12 6 4 4 16 6 18 4 6 10 22 2 20 12 18 6 28 4 30

observation: for bases with simple factorization, the number of graphs and period seem to equal the biggest factor minus one. it also seems that those bases don’t have transient phase. for the bases with multiplicity in their factors (bases 4, 8, 9, 12, 16, 18, 20, 24, 25, 27 and 28), i cannot explain the numbers yet.

seems like i have not been the first human being in this territory, the sequence is apparently known as the Carmichael function, and is referenced to as A002322 in OEIS. still, it’s amazing how far you can go in the depths of maths in a matter of minutes by making a couple of simple questions to youself!

playing around with graphs

i was having a closer look to the other day’s observation about 13² and 31², and i could quickly demonstrate that indeed, for two digit numbers, the only possibilities are 11, 12 and 13. basically, the demo it boils down to the behavior of the last digit when the numbers 0 to 9 get squared.

now this, rather than putting my mind to rest, brought some new questions. see, when doing this squaring number experiment, you can indeed know with which digit your result will end based on the original number. for example, any number ending in 4 will produce a number finished in 6 when squaring. when finished in 3, you’ll get a 9. for 7, a 9 too. for 6, a 6. you can draw a graph that shows how the units of your numbers flow: { 0->0, 1->1, 2->4, 4->6, 5->5, 6->6, 7->9, 8->4, 9->1 }, which has four fixed points. now, the natural question that arises is, what if rather than squaring we start rising the numbers to the third power, or forth, and so on? what kind of patterns/graph will we get? or, how many different graphs will we get?

i couldn’t resist trying out and seeing for myself. this is the experiment:

so, we see than when rising to the cubic power, we get an interesting set of six fixed points and two cycles!! when rising to the forth power, you get something similar to squaring where 1 and 6 become sinks. when fifth powers are computed, however, all the dynamics collapse to 10 fixed points! so it seems we get a rich set of graphs and behaviors. fixed points, sinks, cycles… how many different graphs/dynamics do exist as we increase powers? in theory there are up to 7 source digits landing in 8 destination digits, so that’s at maximum of 2 million potential graphs. do all of them get exist really?

in fact the excitement fades away as soon as we try power 6, which produces the same dynamics as squaring. perhaps just an unfortunate coincidence? not really. seems like power 7 produces the same graph than power 3, and that power 11 and 15 for the matter.

apparently, there are only four different graphs that emerge for all possible power exponents, which are distributed sequentially. how boring…


no me importa desvelarme, o despertarme en medio de la noche,
y pensar que seguramente estoy soñando cuando te creo a mi lado,
y entonces sentir tu calor, escuchar tu respiración, y estirar la mano para tocarte y asegurarme de que no deliro,
y darme cuenta de que estás aquí realmente.

y de las dos posibilidades para lo que continua, la segunda es pegarme a tu espalda, sonreir, y dormirme de nuevo. no me importa despertarme, tantas veces como sean posibles.

beautiful butts

we are walking down polk street. it’s sunny, we have nothing to do for the rest of the day but do nothing, and we feel really good, comfortable in our simple t-shirt and jeans. i am pretty sure hers looks nicer than mines by the way, but still, whoever is walking behind us seems to have his doubts:

– hey guys, i don’t know who of you two has a nicer ass – he says to us

he’s a man, in his 35s, and i’m sure he’s being honest on that he feels attracted to hers as much as he likes mine. this is how this city is after all. and i love it. we laugh, thank him for the double compliment, and wish him a great day.

mirroring squares

i always found it intriguing that 13² = 169 while 31² = 961, and that 12² = 144 while 21² = 441

and of course, same applies to 11² = 121 (and perhaps to 10² = 100 with 01² = 001 too)

give it a try, feels good

although pretty self-consistent for the most part, seems that i am one of those rare people that occasionally change their mind about certain topics, ideologies, positions and believes. it can hurt your pride, but it can help to readjust little things. give it a try, feels good.

my day starts here

i can hear the bell of the cable car from my bed, as it passes by my block. in fact, i just woke up 5 seconds ago, and this is the very first sound i hear today. i smile. then i yawn, stretch out, and sit on my bed. it’s sunny outside, i can see the blue sky through the window. i take a moment to deliberately not think of anything, but listen to my senses instead. comfort, warm, light, colors, sounds, calm. i wait in silence a little bit more, waiting for that magic step to happen, when all these primitive sensations get melted together into one single and higher feeling. happiness. yes, happiness has to be cooked, built and listened to. i learnt to cook happiness long time ago. you just need to allocate time for it. the cable car rings its bell again. i think that i’m lucky that what for most people is no less than the iconic symbol of a great city, it’s nothing but a simple, ephemeral and anecdotal event in my day. the feeling becomes an idea now. i am so lucky. the cable car rings again. okey, gotta go to work, my day starts here.

alternative titles:
– cooking little moments of happiness
– on how sensations lead to feelings, and feelings to ideas, all the way up in a staricase of consciousness

second degree – already crazy complex

For a given parametric parabola of the form

which could well represent a blade of grass growing in direction b and bending down to gravity in direction c, which of course has derivative

the length between its base defined at t=0 and its tip at t=T, is, as we know (after applying the binomial expansion),


which happens to have the not-so-simple solution

Probably T=1, but still, seems that when defined through a velocity and an acceleration, the length of a parabolic shape of a blade of grass is indeed surprisingly complex – it involves a logarithm and a couple of square roots, the point being here that i can’t find an intuitive interpretation to it really. However, we should be grateful there actually exists an analytic solution for it.

Again, we know so little beyond second degree polynomials!

one dollar

i’m walking home late in the dark and lovely night of the city, only interrupted by a few lamps that through some light in the streets. just a few feet in front of me there’s this homeless who is pushing a trolley and comes my way. “nice hat there”. my hat has always been an effective conversation starter, no matter the context. “thx”, i answer. i pay some attention to the guy. he’s in his 50s, has white beard, wears a red cap and i think he has blue eyes. “sorry to disturb. how’s going. can i tell you a joke?”. unlike many homeless, this guy seems to be present and aware, really talking to me. his sight is sharp and focused on me. “sure, go ahead!”. he tells me three variations of the classic “how many XXX does it take to replace a light bulb”. i understand two out of three of the jokes, which is a very unusual score for me, especially given the context (although, again, this guy speaks very clearly). this makes me really happy. he proceeds to explain me that his father is a great joke teller, and that he can remember every single joke he was ever told, even many years after. “i bet you are the same”, i respond, and add a “what’s your name?”. his names is Dave. he asks for my name and he learns my name is Indigo, with d. “okey Dave, i gotta turn right here”. “i’m sorry i am asking this Indigo, but would you by any chance have 75 cents to spare?”. “sure, i have one dollar if that works”. he nods, i give him one dollar, and turn right. “have a good night”, “same for you”.

this random man that i’ll never see again in my life just helped improve my self esteem regarding my english listening skills. and it only costed me a bit of an open attitude, patience and attention. and one dollar. not bad!

at the dentist

– wow, you have two cavities! bad, bad. so tell me inigo, is there anything sweet that you regularly eat???

– hm, other than the 16 spoons of sugar for my cereals every morning, then heavy loaded 6 strawberry jelly toasts of the evening, and the one condensed milk can i have for desert every night for dinner, no, not really. i don’t like chocolate, candies, cookies, cakes or anything like that – i generally don’t like sweet.

– wow. you have ONLY two cavities!

joy in the darkness

it rains heavily in downtown, the water slides fast downhill. the reflection of the neon lights in the wet pavement, the smoke coming out the drains, the ultimate urban environment cliche ever. and the silence of a city that has gone to sleep a while ago, a dark cat hides behind a box in the corner of the alley, the smell of the humid environment. i heading home. but i want to arrive wet today. so i close my umbrella and let the rain drops hit me. i slow down. i’ll take my time.