The M1M1 treasure hunt, week 3

"Before I kill you, Mr Bond," intoned the archvillian, "I shall explain to you my diabolically cunning plan to take over the world, starting with Imperial College Mathematics Dept."

Agent 007 lay strapped to a slab in the villain's lair. He had deliberately allowed himself to be captured, relying on the characteristic weakness of the megalomaniac gloating over him to reveal all his secrets before inexplicably failing to dispose of a helpless adversary.

"You know of course, that the security of all modern communications relies on the difficulty of factorising large integers. Imperial mathematicians have developed a brilliantly simple factorisation algorithm, and they intend to publish their results in the journal of Surprisingly Useful Maths. My spy has discovered that the only copy of this paper lies strewn across the desk in Prof Vulture's office."

"You have a spy in the Mathematics Department?" scoffed Bond. "That is hardly credible - a more incorruptible and selfless collection of benefactors to humanity would be hard to imagine."

"Naturally - my agent believes he or she is working for Amnesty International. Now listen. Recently the College installed a new system of hand-driers in the Huxley lavatories. However, the radiation from these devices interferes with experiments in the Physics Department, and the College now intends to remove the driers. Masquerading as workmen, my men will enter the Huxley building and drill through the walls to Prof Vulture's office and seize the plans. I will then have access to all the computer traffic in the world, which will then be at my mercy."

"That is utterly preposterous - no one would believe that a respected institution such as Imperial would waste vast sums installing and then immediately removing hand-driers," chuckled 007.

"You are mistaken, but that is of no concern. I shall now leave to finalise the details of my fiendish plan. To ensure that you trouble me no longer, I have arranged for this laser to put an end to your meddling. Goodbye, Mr Bond."

007 looked around. A ferociously powerful laser beam was slowly advancing and would shortly bisect his body. The laser casing bore the clearly visible label

"Control Function = cos(π t/2) cos(π t/2 + 1) cos(π t/2 + 2) cos(π t/2 + 3)"

Fortunately, MI6's scientific officer, who was so rational he was known as "Q", had given Bond a watch which could emit a jamming pulse of any frequency. All he had to do was calculate the period of the laser control mechanism as a function of t and type it into the box below. However, he knew that typing the wrong answer would result in instant death.

Period = ?

As luck would have it, the energy burst following the resonant feedback to the laser had melted Bond's bonds, and he was now free. Running up the gangway, he spotted a big red button labelled "Emergency Self-Destruct. Please do not touch." Underneath the button, was the legend "Enter the value of the integral between -1 and 1 of 3x² + sin(x³), and then press the red button. Errors are fatal."

Integral = ?

An alarm sounded. "Termination will occur in 60 seconds." Bond snatched up one of the villain's mobile phones and sprinted towards the exit, aided by the fact that the central administration had, as a cost-cutting exercise, equipped all the guards with blank ammunition.

As debris from the exploding base fell about him, Bond switched on the mobile. Across the screen appeared:

"Options:

(1) sin(x)/x for x ≠ 0 and 1 for x = 0.
(2) x + 2 sin(x).
(3) x/sin(x) for x ≠ 0 and 1 for x = 0.

(a) Odd, increasing and invertible.
(b) Even, continuous and not invertible.
(c) Neither even nor odd, invertible.
(d) Odd, periodic and not invertible.
(e) Even, periodic and not invertible.
(f) Neither even nor odd, discontinuous.
(g) Odd, not periodic, not continuous.

To abort master-plan, enter an integer followed by a letter."

?


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