[Prof. Templeton]

Professor Templeton had a rich but eccentric uncle (aren’t they always). As he got on in years he would think of new and interesting ways to disperse his substantial funds. One afternoon he decided to give away one million dollars to several of his friends, but imposed certain restrictions on himself. He would never give a gift that wasn’t either one dollar or some power of seven. Another stipulation was that seven or more persons would never receive the same amount.

Can you determine how many ways the million might have been handed out and to how many people?

One could always determine the amounts by trial and error, but is there a more elegant way?

[HoustonHokie]

Well, its easy to figure out the minimum number of people the professor's uncle could have given the money to by maximizing the number of high dollar gifts:

7^0 = $1 * 1 = $1

7^1 = $7 * 1 = $7

7^2 = $49 * 3 = $147

7^3 = $343 *3 = $1029

7^4 = $2401 * 3 = $2401

7^5 = $16807 * 3 = $50421

7^6 = $117649 * 1 = $117649

7^7 = $823543 * 1 = $823543

So, the professor's uncle could have given away the million to as few as 16 people.

The only way to give the money to more people would be to remove one of the high dollar gifts and split it up among the lower dollar gifts. The problem is that each high-dollar gift is exactly 7 of the next lower dollar gift, so I'm not sure how the professor's uncle could abide within his rule about not giving the same gift to more than 7 people if he chooses anything but the minimum 16. For example, suppose he wants to give out one fewer $343 gift. The minimum number of gifts to add would be 7 $7 gifts, bring the total number of $7 gifts to 8, which exceeds the maximum. You can try it for any of the amounts, and you'll find it doesn't work.

Perhaps the surest proof is the following:

Try giving 7 gifts each of $1 through $117649 (the maximum in each category). The total is only $960799. You can't give out any more gifts than that, because all categories are exhausted and an $823543 gift would far exceed the one million available.

So I say he has only one option - 16 people in the configuration above.

[Guest]

As 10^6 is 11333311 base 7, the minimum number of people that can receive the million is 1+1+3+...+1=16, where one person each receives $7^7, $7^6, $7 and $1, while three people each receive $7^5, $7^4, $7^3 and $7^2.

Anytime that you wish to split one of these denomination, you must get rid of a person with that denomination and get 7 people to take the next lower power of 7 . The net change in the number of people is always then a multiple of 6.

Clearly, the largest number of people that can receive $10^6 is 10^6, each receiving $1. Note that 10^6=16+6(166664).

Thus, the various numbers of people that can receive money are given by 16+6m, where m is an integer between 0 and 166664.

[Prof. Templeton]

Short but sweet. Well done both.

That one million converted into base 7 is 11333311 which gives us the answer. Keep dividing 1,000,000 by seven until there's nothing left to divide and the remainders of those divisions would be 11333311 (our base 7 number backwards of course).