# Sigma of P(n)

Let’s define a function over the non-negative integers in the following manner:

1. P(n) = n when n is a one-digit integer.
2. P(n) = the product of all the digits of n when n > 9.

Example: P(1729) = 126, because 1 × 7 × 2 × 9 = 126.

Evaluate the following:

Note: this problem was shared to WTM by Reza Kassai, of Shiraz, Iran.

# Arthur’s Arithmogons

Arthur has invented a new puzzle to show his friends. He calls it: Arithmogons.

He draws the following diagram first.

Then he puts any three integers he likes in the circles. In the squares he puts the sum of whatever two integers appear on the ends of the corresponding side.

Before showing this to a friend, he erases the integers in the circles. He tells his friends to find the circle numbers, knowing only those in the boxes.

Here is a sample of Arthur’s Arithmogons for you to solve.

EXTRA: while it’s possible to solve an Arithmogon puzzle by guess-&-check, it would be much more efficient to develop, and prove, a logical step-by-step approach. Such a method exists. Please find it and prove why it works.

# A Quartet of Integers

I have a quartet of integers with which I’ll present a little number problem for you to solve. Here are some facts to work with:

1. Twice the first integer, minus the second, equals 7.
2. Twice the second integer, minus the third, equals 4.
3. Twice the third integer, minus the fourth, equals 3.

Now there are an infinite number of quartets of integers that share those three facts, so I will now add an additional condition:

When you subtract the first integer of my particular group from twice the fourth integer, you get the minimum positive integral value for this operation of all possible cases.

What is the sum of my four integers?

# The Rub-a-dub-dub Restaurant

In Mother Goose City, the most elegant restaurant where the elite meet to eat is the Rub-a-dub-dub Restaurant. It is owned and operated by those three wild and crazy guys: the butcher, the baker, and the candlestick maker. (Perhaps you will recall, they often traveled by tub!)

When they decided to become partners in this expensive project, each promised to contribute as much money as he could, according to the funds he had in his bank account. The results can be described as follows:

1. the ratio of the funds contributed by the butcher and baker was 3 to 5, respectively.
2. the amount of money contributed by the candlestick maker was equal to twice the amount of the butcher less one-half that of the baker.

3. the total amount of money raised by the three investors was greater than 100 mogolas and less than 125 mogolas. (A mogola is an informal unit of money in Mother Goose Land, similar to our use of “grand” to mean \$1000.)

Assuming that each man’s contribution was an integral number of mogolas, how much money did they have to start this culinary adventure?

# Levy Expressions

In July of 2003, WTM had the good fortune to make acquaintance with a very creative mathematician by the name of Jerry Levy. Mr. Levy was attracted by a certain problem based on a popular theme in recreational math, namely pandigitality. Now, pandigitality is a new word not likely to be in the dictionaries yet, as WTM has just invented it here for use in this webpage. At its simplest, it merely means using all the digits 1 to 9 (or in its purest form, 0 to 9) once and only once in some unique manner.

[WTM already has presented some material in this regard. See Digital Diversions and Pandigital Diversions.]

Jerry’s original interest arose from the following situation:

Given the expression

A/BC + D/EF + G/HI

where adjoining letters, such as BC, indicate a 2-place number.

Objective #1: replace the letters with the digits 1 to 9, one digit per letter, in such a way that a integer (i.e. whole number) is the value of the expression.

Objective #2: find the replacements that result in the smallest and largest possible integer values.

It is here where the creative juices in Jerry’s mind began to flow. You see, he then thought to himself, “What if I change the denominator to mean regular algebra notation?” This means that AB now indicates multiplication, as every school algebra student learns. To avoid confusion however, Jerry chose to write the expression this way:

A/(B*C) + D/(E*F) + G/(H*I)

Nonetheless, the two objectives remain the same: to obtain integer values and find the extreme solutions.

Not being content with that interesting situation, he pushed on where no man has gone before, namely these expressions:

(A/B)*C + (D/E)*F + (G/H)*I

(A/B)^C + (D/E)^F + (G/H)^I

Jerry proceeded to work on these problems by hand, or as he wrote in an email, “intuitive trial and error”. Eventually, he made contact with Patrick De Geest, whose website, World!OfNumbers, is a gold mine for number enthusiasts. Soon help was forthcoming from a French school math teacher, Jean-Claude Rosa. Patrick and JC turned their programming skills loose on the problem.

A virtual flood of information soon began to come to light and be offered to WTM. Very interesting it is, too. In fact, so much good data was obtained for the 3rd variant that WTM created a school-level math problem based on it. (See Power-full Fractions.)

No solutions to these problems will be given here in this page. This is a deliberate decision, based on an underlying philosophy of the WTM website, namely to present challenging math materials to the school-aged student, or to the merely curious fan of recreational math, and let them play around with it, making their own discoveries. So dive in and start discovering.

However, what will be given here are some more expression ideas that have been formulated by Jerry, JC, and WTM. (Beware! Some of these cases have not been analyzed by a computer program yet, so proceed at your own risk.)

Variant #4:

A/(B+C) + D/(E+F) + G/(H+I)

Variant #5:

(A+B)/C + (D+E)/F + (G+H)/I

Variant #6:

A/(B-C) + D/(E-F) + G/(H-I)

Variant #7:

(A-B)/C + (D-E)/F + (G-H)/I

Variant #8:

(A + B)*C + (D + E)*F + (G + H)*I

Variant #9:

(A + B)^C + (D + E)^F + (G + H)^I

Variant #10:

(A – B)*C + (D – E)*F + (G – H)*I

Variant #11:

(A – B)^C + (D – E)^F + (G – H)^I

Variant #12:

A^B/C + D^E/F + G^H/I

Variant #13:

A^(B/C) + D^(E/F) + G^(H/I)

Variant #14:

(A + B)*C + (D – E)/F + G^H*I

Variant #15:

(you try one of your own)

What do you do now, you ask? Well, if you like to tinker around with numbers, why not begin by substituting the numbers from 1 to 9 in these expressions, and see if the results are interesting in some way? Like palindromes, primes, etc. What are the largest values, or smallest values, that you can obtain for a given expressiion? How many different distinct values can be found? Let your creativity flow. Numbers are also meant to be enjoyed, in addition to be admired for their utilitarian mode.

# Power-full Fractions

At the start of a new grading period, Ms. Zhong likes to present her pre-algebra class with extra challenging research problems. One time she wrote the following expression on the chalk board:

(A/B)^C + (D/E)^F + (G/H)^I

Then she turned to the class and said: “In this expression you are to replace the variables with the digits from 1 to 9. Then find its simplified value. Your objective, however, is to substitute in such a way that the value turns out to be an integer, that is, a whole number.”

Her students didn’t waste a moment, setting right to work with their paper, pencils, and, yes, their calculators. In no time at all, Andrei raised his hand, and announced proudly, “I’ve got an answer! I found 2283.”

“Very good work, Andrei,” said Ms. Zhong as she examined his calculations.

Just then Trisha said excitedly, “And I’ve got one bigger than Brian’s number.”

After looking at her paper, Ms. Zhong smiled and said, “You certainly do. Nice going.”

From across the room, Timothy, who was often quiet and shy in math class, cautiously proclaimed, “Teacher, I think I have a smaller answer than they do.”

Ms. Zhong went to his desk, patted him on his head as she grinned from ear to ear, saying, “You really do have a nice number there. I’m proud of you all. Now everybody continue. It seems there are lots of different possibilities. This certainly is a power-full problem, isn’t it?”

By the end of the week the class had found quite a long list of integral values. How many can you find?