The CSAT and Practice[+] are designed by the Climb Foundation to help candidates. We are advocates for more opportunity to shine and less opportunity to fail, and we strive to level the playing field. Practice Paper 1 Question 5

Using only the functions $$\max$$ and $$\min$$ and arithmetic operations (no if clauses), express the amount of possible overlap between two intervals $$[a_1,a_2]$$ and $$[b_1,b_2]$$, where $$a_1,a_2,b_1,b_2$$ are arbitrary real numbers with $$a_1<a_2$$ and $$b_1<b_2$$.

• The $$\min$$ and $$\max$$ functions are applied to two or more numbers, and return the smallest and largest of them respectively, e.g. $$\max(-1,-3)=-1.$$

Hints

• Hint 1
Think about intervals on the real number line. Sketch $$[a_1,a_2]$$ and $$[b_1,b_2]$$ when these overlap?
• Hint 2
What is the left boundary of the overlap in terms of min/max of the 4 numbers?
• Hint 3
What about the right boundary?
• Hint 4
What if there is no overlap? Remember you can only use min/max functions.

Solution Assume that intervals overlap, partially or fully, like in the figure above. We can notice that the smallest value in this intersection is always $$\max(a_1, b_1),$$ and the largest value in this intersection is $$\min(a_2,b_2).$$ Hence, the amount of overlap of these intervals can be expressed as $$\min(a_2,b_2)-\max(a_1, b_1).$$

There will be some overlap only if this value is positive, hence the final formula is $$\max(0,\min(a_2,b_2)-\max(a_1, b_1)).$$

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