Number Bases: Level 4 Challenges Practice Problems Online

Let be the number that consists of 61 consecutive 3's, so . Let be the number that consists of 62 consecutive 6's, so . What is the digit sum of ?

Details and assumptions

The digit sum of a number is the sum of all its digits. For example the digit sum of 1123 is .

Let be a polynomial with nonnegative integer coefficients. If and , what is

You have a weighing balance, which you can place weights on both sides of. You need to measure all weights between 1 and 1000.

For example if you have weights 1 and 3, you can measure test objects of weights 1, 3 or 4. You can also measure objects of weight 2, by placing 3 on one side and 1 on the side which contain the object to be weighed.

What is the minimum number of weights that you would need to be able to measure all (integral) weights from 1 kg to 1000 kg?

Calvin went to a newly discovered planet called "Pandora" to research their advancements in the field of Mathematics. He found the following equation scrawled in the dust:

Can you help Calvin in finding the base, which is used in the Number System on Pandora?

Note: Symbols for digits in the Pandora System and Decimal system have the same value. e.g. 6 in Pandora represents six.

Find the number of integers that are between and inclusive that when converted to binary have a digit sum of less than

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Number Bases: Level 4 Challenges

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