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#186 Neural network

Set by Peter Rowlett

Professor Beryl Brainstein is developing a robot called NewRon that uses a so-called neural network to crack open a safe that has a four-digit code. Brainstein gives NewRon two fixed input numbers (she picks 3 and 4) and NewRon uses an algorithm to generate four output numbers (A, B, C and D), which are NewRon’s guess for the safe code.

Each output is connected to the two inputs by arrows labelled with a number called a weight (s, t, u…). NewRon multiplies each input by the weight and adds the resulting numbers to get the output. For example, the inputs 3 and 4 are connected to A by weights s and t, so NewRon’s guess for A will be 3s + 4t. If s = 2 and t = 0.5, then A would be 8. NewRon’s task is to adjust the weights until the output ABCD opens the safe.

Brainstein sets the code for the safe as 1221 and starts NewRon, which whirs for a moment, trying different weights, and… clank, the safe opens. Four of the correct weights are zero and the other four are all different. What are the non-zero weights?

Solution next week

#185 Cheque out

Solution

The original cheque was for £28.64, while the cashier gave Matthew £64.28, so he still has £57.28 to spend. Call the original cheque £a.b. The cashier hands over £b.a, and we know that in pence, 100b+a – 700 = 200a + 2b. Rearranging, you get 98b – 700 = 199a, and we are looking for positive whole number solutions. You can speed things up by spotting that 98 and 700 are both divisible by 14, so 14 (7b – 50) = 199a, which means that 199a is also a multiple of 14, and hence a itself must be a multiple of 14. Quick trial and error reveals that only a=28 gives a whole number for b (=64).