Oh, Digit All!
Credit: Dennis Mojado
1. Easier Than It Looks
In the cryptarithm below, A, B and C are consecutive integers. Solve for their values.
(CB) x (CB) = ACCB
2. Economical Car Trip
You remembered to set the trip meter but didn't look at it until it registered a two-digit number of miles. After a further hour's driving the trip meter read the same two digits, but they had reversed positions. After another hour's driving the trip meter had the same two digits again, this time with a zero between them. If you drove at a uniform speed, what was the first trip meter reading and how fast did you drive?
3. Let's See How Lucky You Are
Pick any positive whole number. Add 5 to it. Multiply it by 21. Now withdraw an even 100, but remember that you did. Multiply what's left by 3. Now repay the 100. Add the digits of you number. If it is a single digit, you are done and you are lucky, yes? If the digit sum is not a single digit, add those digits. Keep adding until the digit sum turns out to be a single digit, and you are still lucky, yes? Can you explain how you got to be so lucky?