Chapter 2

M: That was extra maths

E. Loved it, but quite easy. Here's my problem for you. Hope you know about graphs.

The image is the graph of a function. The positive horizontal and vertical axes are drawn. The graph lies in the positive quadrant.

On the vertical axis there are 26 small horizontal marks, subdividing the vertical axis into equally sized segments.

The function is a polygonal line. There are small filled black circles drawn at points along the graph. Some of these circles also have a second circle drawn around them. The graph is a straight line between the filled black circles.

The graph starts to the right of the 23rd horizontal mark on the vertical axis. This point is marked by a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and down 15 units to a filled black circle.
The graph then moves horizontally across one unit and down 7 units to a filled black circle.
The graph then moves horizontally across one unit and up 19 units to a filled black circle.
The graph then moves horizontally across one unit and down 11 units to a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and up 10 units to a filled black circle.
The graph then moves horizontally across one unit and up 1 unit to a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and down 12 units to a filled black circle.
The graph then moves horizontally across one unit and down 3 units to a filled black circle.
The graph then moves horizontally across one unit and up 4 units to a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and up 5 units to a filled black circle.
The graph then moves horizontally across one unit and up 6 units to a filled black circle.
The graph then moves horizontally across one unit and down 15 units to a filled black circle.
The graph then moves horizontally across one unit and up 13 units to a filled black circle.
The graph then moves horizontally across one unit and down 9 units to a filled black circle.
The graph then moves horizontally across one unit and up 6 units to a filled black circle.
The graph then moves horizontally across one unit and up 3 units to a filled black circle.
The graph then moves horizontally across one unit and down 17 units to a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and up 13 units to a filled black circle.
The graph then moves horizontally across one unit and down 7 units to a filled black circle.
The graph then moves horizontally across one unit and up 5 units to a filled black circle.
The graph then moves horizontally across one unit and down 7 units to a filled black circle.
The graph then moves horizontally across one unit and up 10 units to a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and down 9 units to a filled black circle.
The graph then moves horizontally across one unit and down 5 units to a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and up 17 units to a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and down 13 units to a filled black circle.
The graph then moves horizontally across one unit and up 2 units to a filled black circle.
The graph then moves horizontally across one unit and up 14 units to a filled black circle.
The graph then moves horizontally across one unit and down 9 units to a filled black circle.
The graph then moves horizontally across one unit and down 11 units to a filled black circle.
The graph then moves horizontally across one unit and up 17 units to a filled black circle.
The graph then moves horizontally across one unit and down 14 units to a filled black circle surrounded by a concentric second circle.
The graph then moves horizontally across one unit and up 11 units to a filled black circle.
The graph then moves horizontally across one unit and down 11 units to a filled black circle.
The graph then moves horizontally across one unit and up 1 unit to a filled black circle.
The graph then moves horizontally across one unit and down 2 units to a filled black circle.
The graph then moves horizontally across one unit and down 2 units to a filled black circle.
The graph then moves horizontally across one unit and up 6 units to a filled black circle.
The graph then moves horizontally across one unit and up 8 units to a filled black circle.
The graph then moves horizontally across one unit and down 1 unit to a filled black circle.

The image above contains a hidden question. Please enter your answer to the question in the box below. If the answer is a number please enter it using numerals rather than words.

To deter guessing without thinking, we ask that you also solve the following simple arithmetic problem before checking your answer:

If x = 4, y = 5 and z = 4, what is x + y - z?

This problem was first solved on Mon 20th January at 4:00:22pm

Alan Turing Cryptography Competition 2025 is organised by the The Department of Mathematics at The University of Manchester.
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