Space engineers have long grappled with the problem of how to reliably transmit data from space probes back to earth. How can messages travel hundreds of millions of kilometres without the data becoming hopelessly garbled by noise?

Over the past half-century, mathematicians and computer scientists have devised codes that incorporate redundancy into a message, so that even if noise corrupts some portions, the recipient can usually figure it out. Yet coding theorists have been aware that their codes fall far short of what can, in theory, be achieved.

In 1948, mathematician Claude Shannon, then at the Bell Telephone Laboratories in Murray Hill, New Jersey, published a landmark paper in which he set a specific goal for coding theorists. Shannon showed that at any given noise level, there is an upper limit on the ratio of the information to the redundancy required for accurate transmission. The trouble was that no one could figure out how to construct the super-efficient codes that Shannon’s theory had predicted. In the mid-1990s a pair of French engineers astonished the insiders with their invention of what they called turbo codes, which come within a hair’s breadth of Shannon’s limit. Later in the decade, coding theorists realised that a long-forgotten method called low-density parity-check (LDPC) coding could get even closer to the limit.

PUZZLE 1: I was sitting around with my friend Waldo, his nephew Spike, and Spike’s friend Molly recently. I happened to have two tickets to a new movie in my pocket that I had just purchased, and I mentioned this and noted that there were two four-digit numbers on the tickets and that the sum of all eight digits was 25. Waldo asked if any digit appeared more than twice out of the eight, which I answered, and then Spike asked if the sum of the digits of either ticket was equal to 13, which I answered also. Much to my surprise Molly immediately told me what the two numbers were. What were they?

Solutions on December 5

CORRECT ENTRIES

November 7

Ravi Raja,Calcutta ? 20; Debasis Ganguly, Alumnus Software Ltd; Pritam Bhattacharya, Calcutta - 94; Sidharth Udani, MP Birla Foundation Higher Secondary School; Abhishek Dey Das, Barrackpore; Tirtharaj Bhaumik,Texas A&M University; Arnab Kr Sadhukhan, CMCC-Jadavpur; Sanjeev Kedia, Bangur; Debanshu Sinha and Shouvik Sarkar, Jamshedpur;

CORRECT ENTRIES

October 31

Sandeep Garg, Howrah; Debjani Ghosh, Rabindra Nagar, Behala, Abhishek Agarwal, Guwahati; G. Srinivas Chakradharpur; Dhruba Jyoti Daityari, Calcutta - 75; S.P.S. Jain, New Delhi; D.R. Subrato, KITS-Bhubaneswar; S. K. Choudhary, Durgpur Please send your entries to knowhow@abpmail.com within 10 days. Send complete solutions, not one-line answers.

PUZZLE CRACKED

Solution 1: I am Molly. Let Waldo start with w, Molly with m, and Spike with s. Following the above account we get the following progression of monies: W=2w, M=m-w; M=2(m-w), S= s-(m-w)=s+w-m; S=2(s+w-m), W=2w-(s+w-m) =w+m-s. And so Waldo finished with w+m-s, Molly with 2(m-w), and Spike with 2(s+w-m). Since these must all be equal, we have three equations and three unknowns, so solve; this gives us that 4m=5s and 3s=4w. Now, if s=1/2, this implies that w=3/8=37.5 cents, an impossible amount of money to start with. If w=1/2, this implies that s=2/3, again an impossible amount to start with. Finally, if m=1/2, this implies that s=2/5=40 cents, w=3/10= 30 cents, which works. Solution 2: The pyramid can indeed be built in 53 days. These puzzles appeared on November 7