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Today Puzzle #748

Puzzle No. 748鈥� Wednesday 27 May

A large café is preparing to re-open post lockdown. Each table for a household is 1 square metre in area. In order to preserve social distancing the edge of one table can’t be closer than 3m to another. Should the owner use square or circular tables to maximise the number of tables?

Today’s #PuzzleForToday has been set by Dr Shaun Fitzgerald, Fellow of Girton College

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The square tables can be arranged by positioning them so that there is 3m distance from the corner of a table to the next, and where two of the side edges (opposite ones) are also 3m from an adjacent table. The area of a rectangle formed with opposite corners at the centre of two tables is 2 x (1+SQRT(8)) and it contains half a table. The area occupied by a whole table is therefore 2 x 2 x (1+SQRT(8))=15.3m2.

The circular tables need to be arranged in close packed circles, each with 1.5m distance added to the radius r of the table. The radius r of a 1m2 table is given by r=sqrt(A/p) = sqrt(1/p) = 0.56m. Consider the smallest triangle formed from the centres of three adjacent tables. The base b of the triangle is 3+2r and the area of the triangle is 0.5 x b2 x sqrt(3)/2. The triangle contains half a table so the area occupied by one full table is 2 x 0.5 x b2 x sqrt(3)/2 = 14.8m2.

This is less than the area used by each square table and therefore the circular ones should be used.

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