Sunday Afternoon Maths LX
Where is Evariste?
Evariste is standing in a rectangular formation, in which everyone is lined up in rows and columns. There are 175 people in all the rows in front of Evariste and 400 in the rows behind him. There are 312 in the columns to his left and 264 in the columns to his right.
In which row and column is Evariste standing?
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The common factors of 175 and 400 are 1, 5, 25. Hence in front and behind him there are either:
| In front | Behind |
| 175 rows of 1 | 400 rows of 1 |
| 35 rows of 5 | 80 rows of 5 |
| 7 rows of 25 | 16 rows of 25 |
Therefore the shape of the formation is either 716×1, 126×5, or 24×25.
The length of the columns (716, 126, or 24) must be a factor of 312 and 264. Therefore it is 24.
Evariste is standing in the 8th row from the front and the 14th column from the left.
Extension
Evariste is standing in a rectangular formation, in which everyone is lined up in rows and columns. There are \(a\) people in all the rows in front of Evariste and \(b\) in the rows behind him. There are \(c\) in the columns to his left and \(d\) in the columns to his right.
For which numbers \(a\), \(b\), \(c\) and \(d\) is the row and column in which Evariste standing uniquely determined?
Bending a straw
Two points along a drinking straw are picked at random. The straw is then bent at these points. What is the probability that the two ends meet up to make a triangle?
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A triangle will be made if none of the segments of straw is longer than the other two added together. This is the same as requiring that each segment must be less than half the straw.
Let the length of the straw be 1 unit. Call the points \(x\) and \(y\). A triangle is made if either:
- \(x\lt y\), \(x\lt\tfrac12\), \(y-x\lt\tfrac12\), \(1-y\lt\tfrac12\); or
- \(y\lt x\), \(y\lt\tfrac12\), \(x-y\lt\tfrac12\), \(1-x\lt\tfrac12\).
For the second condition, the allowable region is shown below.
This region covers \(\tfrac18\) of the whole square. By switching \(x\) and \(y\) it can be seen that the first condition's region is the same size as the second's, plus they don't overlap. Therefore the probability of making a triangle is \(\tfrac18+\tfrac18=\tfrac14\).
Extension
One point along a drinking straw is picked, then a coin is flipped. If the coin shows heads, a second point above the first is chosen; If tails, a second point below the first is chosen. The straw is then bent at these points. What is the probability that the two ends meet up to make a triangle?
![](javascript:showlimage("straws_1.png"))