Striker caroms
Algebra
Level
4
A boy has three carom strikers ( take them as disc) , each of radius 1 . He wants to fit them in one of his boxes side by side i.e non-overlapping one another. Find the length of the side of smallest square box he may use for this purpose.
The answer is 3.9319.
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1 solution
Let discs have centres P, Q and R and square is named STUV. Firstly, in the smallest square containing all 3 discs , a pair of parallel sides must each be tangent to a disc. If these sides are to called side 1 and 2 then we can slide the square ( if necessary) to bring a side 3 into contact with a disc. If these three sides meet three different discs,we can, if necessary, slide the square so that the fourth side instead meets the disc. This must be a different disc to the one meeting side 3 ( for the square containing the 3 discs in a line is not minimal , its diagonal must be \geq 6 and area \geq (6/√2)^2 = 18. Hence we may assume that there is a disc which touches a pair of adjacent sides. Say disc 1 meets sides named TS, SU. It follows that SP is on the diagonal SV of the square. Now PQ,QR and RP all have length at least 2 . Hence if any of these makes an angle \geq π/12 with a side , then the side is longer than 1+1+2cosπ/12 = (4+√2+√6)/2