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#187 Key decision

Set by Howard Williams

It is a big day for Mr Hopeful. He has got through to the last stage of a television game show, with the chance of winning a car. If he chooses the correct casket from the four on offer, he will find the keys inside and be driving home in a brand new, top-of-the range electric vehicle.

He is confident, as he knows that each of the caskets has a clue.

However, his self-assurance is shattered when the game show host explains that three of the four clues are false.

Can you help Mr Hopeful decide which casket to choose to win the car?

Solution next week

#186 Neural network

Solution

There are several equivalent answers, all with half the weights zero and the other half 1/4, 1/3, 1/2 and 2/3.

To get A = D = 1 we must have 3s + 4t = 1 and 3y + 4z = 1. If we have both weights into one neuron as zero, that neuron will be zero. Since all the outputs are 1 or 2 and half the weights are zero, it must be that each neuron has one weight zero and one non-zero. Choosing s and z to be zero gives t = 1/4 and y = 1/3. In the same way, with B = C = 2 we have 3u + 4v = 2 and 3w + 4x = 2, which can be solved with u and x as zero, using v =1/2 and w =2/3.