Extra output layer in a neural network (Decimal to binary)
The question is asking you to make the following mapping between old representation and new representation:

Represent    Old                     New
0            1 0 0 0 0 0 0 0 0 0     0 0 0 0 
1            0 1 0 0 0 0 0 0 0 0     0 0 0 1 
2            0 0 1 0 0 0 0 0 0 0     0 0 1 0 

3            0 0 0 1 0 0 0 0 0 0     0 0 1 1 
4            0 0 0 0 1 0 0 0 0 0     0 1 0 0 
5            0 0 0 0 0 1 0 0 0 0     0 1 0 1 

6            0 0 0 0 0 0 1 0 0 0     0 1 1 0 
7            0 0 0 0 0 0 0 1 0 0     0 1 1 1 
8            0 0 0 0 0 0 0 0 1 0     1 0 0 0 

9            0 0 0 0 0 0 0 0 0 1     1 0 0 1


Because the old output layer has a simple form, this is quite easy to achieve. Each output neuron should have a positive weight between itself and output neurons which should be on to represent it, and a negative weight between itself and output neurons that should be off. The values should combine to be large enough to cleanly switch on or off, so I would use largish weights, such as +10 and -10.

If you have sigmoid activations here, the bias is not that relevant. You just want to simply saturate each neuron towards on or off. The question has allowed you to assume very clear signals in the old output layer.

So taking example of representing a 3 and using zero-indexing for the neurons in the order I am showing them (these options are not set in the question), I might have weights going from activation of old output $i=3$, $A_3^{Old}$ to logit of new outputs $Z_j^{New}$, where $Z_j^{New} = \Sigma_{i=0}^{i=9} W_{ij} * A_i^{Old}$ as follows:

$$W_{3,0} = -10$$
$$W_{3,1} = -10$$
$$W_{3,2} = +10$$
$$W_{3,3} = +10$$

This should clearly produce close to 0 0 1 1 output when only the old output layer's neuron representing a "3" is active. In the question, you can assume 0.99 activation of one neuron and &lt;0.01 for competing ones in the old layer. So, if you use the same magnitude of weights throughout, then relatively small values coming from +-0.1 (0.01 * 10) from the other old layer activation values will not seriously affect the +-9.9 value, and the outputs in the new layer will be saturated at very close to either 0 or 1.