Challenge:
The famous cooking gnome Oliver James made three of his delicious Christmas puddings for the elves’ Christmas party.
The stuffy elf Symmetrix is pretty annoyed at the sight of the three puddings and notes grumpily: “Was is not possible for this bungler to produce three perfectly alike puddings of the same size?!”
Elf Geometria is however very delighted: “Look how beautifully arranged these Christmas puddings are: all three are shaped exactly like hemispheres. Each one of them is placed onto its flat side with its vault directed towards the ceiling. The lower two puddings sit side by side on a flat plane, and they touch at exactly one point. They are of different size; however, the upper pudding, which is placed onto the lower two, has exactly the right size such that its diameter conincides with the distance of the contact points of the lower puddings with their mutual tangent plane.” See Figure 1.
We want to know: If the bigger of the two lower Christmas puddings is of volume , and the smaller one is of volume , what is the first digit of the fractional part of the volume of the upper Christmas pudding?
Artwork: Frauke Jansen
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Comment:
Catriona Shearer, the author of this challenge, is a math teacher at a secondary school in Cambridge, UK. As @Cshearer41, she posts a new geometric puzzle every day on Twitter. Warning: Highly addictive!