Is a Neural network with 2 input nodes, 2 hidden nodes and an output supposed to be able to solve the XOR problem provided there is no bias? Or can it get stuck?

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There are more limitations to not having a bias. It's not possible to have XOR without a bias.

You cannot solve XOR without having a bias neuron, as it would require that you increase the number of hidden neurons to a minimum of 3 hidden neurons. In that case, the 3rd neuron basically acts as a bias neuron.

If you wish to know about Neural Network then visit this Neural Network Tutorial.