Intermediate Value Theorem (IVT)

$$\text{Trained Detective}$$

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This puzzle is a tribute to Murder on the Orient Express, one of Agatha Christie's most famous books.

Mr. Fletcher has been shot on the Orient Express! Coincidentally, the world's master detective (you) are on the train. You narrow down the list to five suspects who had previous dealings with him: Mr. Andrews, Ms. Berryman, Colonel Clifton, Dr. Dunning, and Ms. Eddy. You interviewed them separately, and specifically ask them to detail every encounter they had with each other the other suspects that night.

Mr. Andrews:

I witnessed a fiery dispute between Colonel Clifton and Mr. Fletcher while we were in Fletcher's cabin. As the train left Budapest, Clifton and I left the cabin, and Mr. Fletcher stayed behind. Since Clifton was in a foul mood, I went the other way, to the restaurant car at the back of the train for a leisurely supper. Ms. Berryman passed me in the opposite direction.

Ms. Berryman:

Ms. Eddy treated me to dinner, signing the bill to charge it to her cabin. We parted ways in opposite directions, and I remember passing a very uncomfortable Mr. Andrews. I continued onward to my cabin, to retire for the evening.

Colonel Clifton:

Mr. Andrews and I visited Mr. Fletcher to settle some business accounts, where we had a disagreement. His cabin was uncomfortably hot, so I left around 8pm. I didn't have an appetite, and walked to the corridor of the next car forward to cool off. Not long after, Dr. Dunning entered in apparent hurry, and she picked up something from the floor before passing through to Mr. Fletcher's car.

Dr. Dunning:

My neighbor Ms. Eddy knocked on my cabin, asking for some sleeping pills. I went out to find water. Along the way, I passed Colonel Clifton, and noticed Mr. Fletcher's watch on the ground. I noticed that it was half-past-eight. I took it to his cabin. My knock opened the door to a cold gust of wind, and I discovered that he had been shot!

Ms. Eddy

I was feeling a bit sick after dinner with Ms. Berryman, who confided in me about the nature of her past relationship with Mr. Fletcher. I asked Dr. Dunning for medical assistance. Dr. Dunning walked off in the direction of Mr. Fletcher's car, and did not return.