Thue showed that there exist arbitrarily long square-free strings over an alphabet of three symbols (not true for two symbols). An open problem was posed, which is a generalization of Thue’s original result: given an alphabet list L = L1, . . . , Ln, where |Li| = 3, is it always possible to find a square-free string, w = w1w2 . . . wn, where wi ∈ Li? In this paper we show that squares can be forced on square-free strings over alphabet lists iff a suffix of the square-free string conforms to a pattern which we term as an offending suffix. We also prove properties of offending suffixes. However, the problem remains tantalizingly open.
Thue showed that there exist arbitrarily long square-free strings over an alphabet of three symbols (not true for two symbols). An open problem was posed, which is a generalization of Thue’s original result: given an alphabet list L = L1, . . . , Ln, where |Li| = 3, is it always possible to find a square-free string, w = w1w2 . . . wn, where wi ∈ Li? In this paper we show that squares can be forced on square-free strings over alphabet lists iff a suffix of the square-free string conforms to a pattern which we term as an offending suffix. We also prove properties of offending suffixes. However, the problem remains tantalizingly open.
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1. Learning target:
• Graph proportional relationships, interpreting the unit rate as the slope
of the graph.
• Compare two different proportional relationships represented in
different ways. For example, compare a distance-time graph to a
distance-time equation to determine which of two moving objects has
greater speed.
2. Alex skateboards at a constant speed from his house to school 3.8 miles away. It
takes him 18 minutes.
a. What fraction represents his constant speed, C?
b. After school, Alex skateboards at the same constant speed to his friend’s
house. It takes him 10 minutes. Write the fraction that represents constant
speed, C, if he travels a distance of y.
c. Write the fractions from part (a) and (b) as a proportion, and solve to find
out how many miles Alex’s friend’s house is from school. Round your answer
to the tenths place.
3. Vicky reads at a constant rate. She can read 5 pages in 9 minutes. We want to
know how many pages, p, Vicky can read after t minutes.
a. Write a linear equation in two variables that represents the number of pages
Vicky reads in any given time interval.
4. Vicky reads at a constant rate. She can read 5 pages in 9 minutes. We want to
know how many pages, p, Vicky can read after t minutes.
b. Complete the table below. Use a calculator and round answers to the tenths
place.
5. Vicky reads at a constant rate. She can read 5 pages in 9 minutes. We want to
know how many pages, p, Vicky can read after t minutes.
c. About how long would it take Vicky to read 25 pages? Explain.
6. Is the point (1, 3) a solution to the linear equation 5x − 9y = 32? Explain.
7. Find three solutions for the linear equation 4x − 3y = 1, and plot the solutions
as points on a coordinate plane
8. Ethan found solutions to the linear equation 3x − y = 8 and graphed them. What
shape is the graph of the linear equation taking?
14. Explain why the graph of a linear
equation in the form of y = c is the
horizontal line, parallel to the x-axis
passing through the point (0, c).
x y
15. Explain why there is only one line with
the equation y = c that passes through
the point (0, c).
x y
