CH. XII] MISCELLANEOUS PROBLEMS 255 



which is put down. If three of the discs are placed so that their 

 edges pass through 0, the radius of each of them must exceed 

 •6099578 foot, a length practically indistinguishable from that of 

 the radius of the minimum disc. If the discs are put with their 

 centres at the vertices of an inscribed pentagon and their edges 

 passing through 0, the radius of each of them must exceed 

 ■6180340foot. It follows that unless the discs are cut with extreme 

 accuracy the problem may be solved by making the circumference 

 of each disc pass through 0; the possibility of using this 

 inaccurate rule is a serious defect in the problem when used as 

 the foundation of a puzzle. 



I believe that the discs used in fairs are generally large 

 enough to allow of the employment of the inaccurate rule, 

 though even then it is safer to use the correct method. In an 

 example made for myself I put a minute faint mark near the 

 centre of the red circle but just far enough away to ensure failure 

 for those who make that point lie on the edge of each disc. 

 Notwithstanding their neglect or ignorance of the correct rule 

 showmen seem to find that the game is profitable, and obviously 

 this is an excellent test of its merits from their point of view. 



Tesselation with Super-Dominoes. A novel form of 

 tesselation problem was suggested in 1921 by Major MacMahon* 

 The object is to fill, according to certain rules, a prescribed area 

 with wood or cardboard super-dominoes. The shape of an ordi- 

 nary domino is a rectangle, the breadth of which is immaterial, 

 with two ends or faces on which numbers or pips can be inscribed, 

 and if we like each number may be taken to indicate a particular 

 colour; by using n numbers we can get n(w + l)/2 different 

 linear dominoes. If our dominoes are triangular in shape, and 

 from the centres of the triangles we draw lines to the angular 

 points, we get dominoes with three faces on each of which num- 

 bers or pips can be inscribed, and as before each number may 

 be taken to indicate a particular colour: by using n colours we 

 can get n (n 2 + 2)/3 different triangular dominoes. Similarly, by 

 using n colours we can get n (n + 1) (w 2 — n + 2)/4 different square 



* F. A. MacMahon, New Mathematical Pastimes, Cambridge, 1921. 



