Can we design a game that a computer cannot play well?
There already Is, at least one 'board game' that has been accompanied by a description 'online' as, Not currently 'solvable-by-computers' ..{the name, of which escapes me} - But, generally speaking - Designing, such a game, Only requires something, that 'humans' Still have, a decided 'edge' at - Namely - Imagination! o:

this is not true. only because we limit competition to humans, but of course a car can run faster than Usain Bolt and a speed boat can swim faster than Michael Phelps. in future robots will be superior to humans in all sports - football, basketball, etc.
Cars and boats aren't computers. They also aren't playing the sport.
Now that they can essentially be made to learn from mistakes (although I don't completely understand alpha go's learning) and because humans program computers (so whatever we come up with, we should be able to express in programming language as well) I doubt it.
Well, of course we can design something nothing can play well, but if you mean humans can play it well and computers cannot, then I don't think so.
Cars and boats aren't computers. They also aren't playing the sport.
you are looking at this in a narrow view.
cars and boats are machines which are the first stage in the evolution of computers. yes they are not designed to play any sports but their purpose is fast transportation, in which case they are capable of moving at faster speeds than any human has the physical capacity to do.
just think of what robots in the future can do because they are not limited by biological constraints. they will be able to play sports one hundred percent perfectly without ever getting tired or distracted, always performing at peak condition, never making mistakes. combine that with elements of AI such as alertness, awareness of surroundings, super-human reflexes/precision and strength, and decision making abilities that are optimized to produce the highest probability of winning each and every time (e.g., passing to teammates over taking a shot yourself, etc.)
computers have been doing all this in intellectual fields for a long time. and there is no reason to not believe that robots will someday become the best competitors, both mentally and physically, in all human sports.
It's already trivially easy to make a team of robots that will beat any human team at any sport... but having a giant box on wheels isn't sporting or interesting.
What we haven't done is make anthropomorphic, bipedal, human sized... i.e. androids that play sports.

Any games that bots aren't already beating are down to either the lack of bots in the game in question, or the competence of the programmer in either his/her knowledge of the game or his programming abilities.
In recent years I have noticed a number of bots that acquire their skill from the abilities of the programmer through multiple simulations and training the bot to learn from its results. There is a limit hold'em bot that used this method to develop an approximation of a gto strategy in poker, and it sounds as though this is what alpha go did as well.
The current engines that we have in chess, I would argue have gained their success through the decision tree pruning skills of the programmers that allow them to look deeper, whereas developments in the evaluations of positions for engines does not seem to have much of a noticeable effect. In contrast to the bots in poker/go, I am unaware of a bot in chess that has used multiple simulations of games to learn and develop a strategy, although I still believe it is predominantly the skill of the programmer that makes komodo the best player right now.
I feel this is probably due to the fact that most humans that have become successful at one game or another, either do not have the programming knowledge required to create a bot, or do not want to see bots in their profession, for example bots in poker have a negative impact on the game. If a top player at chess/poker/go wanted to create a bot, I am sure they would have no problems making their bot one of the best in the world at their game, if they were given the programming knowledge required to create one.

Regarding poker bots. I know it has already happened and it has worked.
Bots which are very good in online poker. [includes no limit poker]
Thus you will be seeing poker pros gradually switching more from on line poker to more of "in person" poker.
[or I should say poker pros setting up the online bots while playing "in person" more often]

The point I was trying to make is that a decent programmer should be able to make a bot beat any thinking game. I went on a bit of a tangent maybe because I've thought about this subject in depth before.

In Bridge signalling your partner non-verbally is against the rules, but in some cases your partner can read your body language.
Sorry but have to disagree with you here. In bridge it is very common to signal your partner to something just by what card you play.
[i play only on internet where you cannot see your human or robot partner]

bots in money bridge on the internet can be maddening. Sometimes they play like a grandmaster and sometimes they play like a class D player.
Remember one hand in money bridge where my partner [a bot] and I bid very nicely to 7 Clubs which would have made for substantial money. But then my bot partner went on to 7 No Trump which did not have a chance to make.
Not sure, if this was already 'covered'!? .. And, Yes.. I Am, 'Lazy'!
https://en.wikipedia.org/wiki/Arimaa
Duplicate bridge is a game computers play poorly. In duplicate bridge, computers are sometimes called "bots" I have played hundreds of games with a computer as a partner and am sorry to say that the best computer is only equivalent to a Class A player in chess.
I think it is just because the dupicate bridge programmers are not all that good at duplicate bridge themselves?