#53 Paintings by numbers
When the famous artist Pablo Picossa held his final exhibition at the Galleria del Pardo, he wanted the public to experience his works in the order in which he had created them. Paintings from his early 鈥淕reen鈥� period were in room 1. From there, visitors should go to room 2 to see his Mauve works and then to the adjacent rooms 3, 4, 5 and so on, until they reached the Black paintings (generally viewed as Picossa鈥檚 darkest period) in room 9.
Alas, no details remain to indicate which room was where. Yet his widow Bella does recall a curiosity about the numbering of the rooms: the three-digit number formed by the top row added to the number formed by the middle row equals the number formed by the bottom row. Can you recreate Picossa鈥檚 gallery tour?
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#52 Bus change
Solution
The easiest way to work out the maximum amount of change you can have without having 拢1 exactly is to start from the largest coins and work your way down.
You can鈥檛 have 拢1 or 拢2 coins, but you can have a 50 pence coin. Then you can add up to four 20p coins and still be unable to make 拢1.
From here, you can鈥檛 add any 10ps (or you鈥檇 have 50p + 20p + 20p + 10p = 拢1), but you can play the same trick again with 5ps and 2ps, adding in one 5p and four 2ps. 50 + (20 x 4) + 5 + (2 x 4) = 拢1.43 in total.
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Quick quiz #45
1 Which mathematician and astronomer made a conjecture in 1611 about how closely you can pack spheres in a 3D space?
2 According to the idea, about how much of a space can you fill with spheres of the same size: two-thirds, three-quarters or four-fifths?
3 When was this considered formally proved, using programs by Thomas Hales and collaborators, in a paper in Forum of Mathematics?
4 Mathematicians have tried to solve this problem in higher dimensions. Does the maximum amount of space you can fill go up or down?
5 Which 鈥渇ather of information theory鈥� showed that this higher-dimensional mathematics was useful in reconstructing noisy communications signals?
Answers below
Quick quiz #45
Answers
1 Johannes Kepler
2 About three-quarters, 74 per cent. Experiments show you only achieve about 65 per cent if you drop them in at random
3 2017
4 Down, precipitously: in 7D, for example, it seems to be only around 30 per cent
5 Claude Shannon, with the sampling theorem that often bears his name