Numerical Logic Puzzles

7/28/2019 Numerical Logic Puzzles

1/18

NUMERICALLOGICPUZZLES--- TRUE/FALSEDesignedby RobertA. Kraus

Backto LOGICPUZZLES

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To solve the puzzles you should have fairly good ability and knowledge of basic logic, high-school algebra, and the

simplest ideas of elementary number theory (e,g, primes, squares, divisibilty, etc.). Most of the problems require the

compilation and sorting of large lists of numbers and the use of a good spreadsheet program is necessary. The type and

difficulty of the problems varies, some are quite long and difficult.

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The Council of Numeria

On the island of Numeria each of the natives is one of two types: Truth-Tellers who always tell the truth, or Liars

who never tell the truth. The island is governed by a Council of Elders who will only answer questions that have numerical

answers. In fact the only answers they give are whole numbers, either zero or positive. Furthermore, they will never give an

answer greater than the current number of council members. This number can vary daily, but is never less than 4 or more

than 40. Also, the Council will only answer questions whose correct answer is independent of who is asked (e.g., no

questions such as "How old are you?").

One day three native students, Ann, Bob, and Cal, were given an assignment by their teacher to question the

council. They each asked a question, which was answered by every council member. Afterward they reported to their

teacher and made the following statements:

(1) Ann: I asked the council how many of them were Truth-Tellers.

(2) Bob: I asked the council how many of them were Liars.(3) Cal: Those statements are not both true!

(4) Ann: All of the answers I received were different (no two equal to each other).

(5) Bob: All of the answers I received were different (no two equal to each other).

(6) Cal: At least two of my answers were different (not equal to each other).

(7) Ann: The sum of my answers is a palindrome.

(8) Bob: The sum of my answers is a palindrome.

(9) Cal: The square root of the sum of my answers is not less than the number of council members.

What was the number of council members on that day?

View Solution

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2/18

The Islandof Brightsand Braves

On a remote island all of the natives belong to one of two tribes: the Brights, who are so brilliant at numerical

calculations that they always get the correct answer, and the Braves, who bravely rush in to do calculations beyond their

ability and never get the right answer. (The Braves are not entirely stupid: they can do simple counting and comparing of

numbers, but they always get arithmetic calculations wrong.) Both Brights and Braves pride themselves on their

complete honesty. They always tell the truth, or (in the case of Braves) at least what they believe to be the truth; theynever purposely tell a lie (unlike the folks on some of those other islands).

One day a group of natives was playing a game of Numberskulls. There were 5 players and a moderator. The

moderator, who was a Bright, painted a 3-digit number on each of the players' foreheads, so that each could see all

numbers but their own. All 5 of the numbers were different. The moderator would ask them questions in turn about the

numbers they could see, and from the answers they would try to deduce what number was on their own forehead. The

first to do so was the winner. What follows is a record of the game, with questions omitted and players designated by

letters.

(1) A: I see exactly 1 prime number.

(2) B: I see exactly 2 prime numbers.

(3) C: I see exactly 3 perfect squares.

(4) D: I see exactly 3 triangular numbers.(5) E: I see exactly 3 perfect squares.

(6) A: I see exactly 3 numbers with a digital sum of 10.

(7) B: I see exactly 3 numbers whose square root is more than 25.

(8) C: I see exactly 0 numbers with a digital sum of 10.

(9) D: I see exactly 3 perfect cubes.

(10) E: I see exactly 0 numbers with a digital product of 18.

At this point one of the players announced his number and won. (Of course it was a Bright; for some reason Braves

never win these games, a point of much amusement to the Brights!)

What number was on each player's forehead?

View Solution

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3/18

The Trial of the Gods

In the ancient land of Num-Burr existed a secret society whose beliefs were a strange mix of mathematics, logic, and

mysticism. Potential new members had to undergo the Trial of the Gods.

Six of the Elders met with the applicant and each Elder made five statements, of which four were true and one was false.

The statements referred to the Divine Qualities, the Sacred Numbers, and the Number of the Gods. The applicant had to

logically determine the Number of the Gods. The applicant was confined to a prison cell with water, light, and writingmaterials. After three days he was brought before the Elders and asked for the Number of the Gods. If he answered

correctly he was admitted to the society; if not he was executed. Here is a record of the statements, with the Elders

designated by letters, and all Numbers are positve integers:

(A1) No two Sacred Numbers are equal.

(A2) The sum of the Sacred Numbers is not a perfect cube.

(A3) The Number of the Gods is a perfect cube.

(A4) All Sacred Numbers have the same Digital Sum.

(A5) Each Sacred Number has at least 2 digits but less than 5 digits.

(B1) At least 2 of the Sacred Numbers have the Divine Quality of being Palindromes.

(B2) At least 2 of the Sacred Numbers have the Divine Quality of having their digits in Strictly Increasing order (no 2digits equal).

(B3) At least 3 of the Sacred Numbers have the Divine Quality of having their digits in Strictly Decreasing order (no 2

digits equal).

(B4) The total number of digits of all of the Sacred Numbers is less than 22.

(B5) All of the Sacred Numbers have Digital Sums greater than 18.

(C1) At least 1 of the Sacred Numbers has the Divine Quality of being Square.

(C2) At least 3 of the Sacred Numbers have the Divine Quality of being Triangular.

(C3) At least 3 of the Sacred Numbers have the Divine Quality of being Prime.

(C4) At least 2 of the Sacred Numbers possess 3 of the 6 Divine Qualities.

(C5) None of the Sacred Numbers has 4 digits.

(D1) The number of Sacred Numbers is not Square.(D2) The number of Sacred Numbers is not Triangular.

(D3) The number of Sacred Numbers is not a perfect cube.

(D4) The sum of the Sacred Numbers is a perfect cube.

(D5) At least one of the Sacred Numbers has 2 digits.

(E1) No two Sacred Numbers have the same Digital Product.

(E2) Exactly 1 Sacred Number has a Digital Product which is Prime.

(E3) Exactly 1 Sacred Number has a Digital Product which is Triangular.

(E4) Exactly 3 Sacred Numbers have Digital Products which are Square.

(E5) No Sacred Number has a Digital Product which is less than 8.

(F1) Each Sacred Number has at least 3 digits.

(F2) The Number of the Gods is not a perfect cube.

(F3) The sum of the Sacred Numbers and the Number of the Gods are not both perfect cubes.

(F4) The Number of the Gods possesses none of the 6 Divine Qualities.

(F5) The Number of the Gods is equal to the sum of the Digital Products of all of the Sacred Numbers.

What was the Number of the Gods?

View Solution

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4/18

Digital Dilemma

Let N be a positive integer, base 10.

Let D be the number of digits of N.

Let A be the number of digits of N equal to 1.

Let B be the number of digits of N greater than 1.Let C be the number of digits of N equal to 0.

Let S be the sum of the digits of N.

Let P be the product of the digits of N.

Below are 5 sets of 4 statements each concerning N. Half of the statements are true and half are false. There are a

different number of true statements in each set.

(1-1) N is a Palindrome.

(1-2) B = C

(1-3) A > 4

(1-4) B is Odd.

(2-1) P is a Prime.

(2-2) A > B

(2-3) All of the digits of N are different.

(2-4) No digit of N is greater than the next following digit.

(3-1) N is not a Palindrome.

(3-2) B > 2

(3-3) A is Even.

(3-4) All of the digits of N are the same.

(4-1) D is a Prime.

(4-2) No digit of N is less than the next following digit.

(4-3) The first digit of N is greater than 2.(4-4) P = S

(5-1) A equals the product of exactly 2 different primes.

(5-2) B > 3

(5-3) A is divisible by a square > 1.

(5-4) S is not a Square.

Find the value of N. (You might want to express it in descriptive or arithmetic form!)

View Solution

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5/18

Digital Logic

Let N be a positive integer, base 10.

Let D be the number of digits of N.

Let S be the sum of the digits of N.

Let E be the value of the smallest non-zero digit of N.

Let K be the value of the largest digit of N.Note: This does NOT exclude the possibility that K=E.

Let M be the number of digit-values appearing in N which lie strictly between K and E.

Let A be the number of times E appears as a digit of N.

Let B be the number of times K appears as a digit of N.

Let C be the number of times zero appears as a digit of N.

Below are 4 sets of statements.

Each set contains exactly 2 false statements, the rest being true.

(1-1) N is a Palindrome.

(1-2) S is a Prime.

(1-3) There are exactly 4 pairs of unequal adjacent digits in N.(1-4) M = 0

(1-5) (A+B) is a Square.

(1-6) The last digit of N is 0.

(1-7) N contains more than 7 different digital-values.

(2-1) D < 1000

(2-2) C = M

(2-3) C is a Square.

(2-4) D is a Square.

(2-5) The first digit of N is the same as the last digit of N.

(3-1) S = D

(3-2) K = E(3-3) K is a Prime.

(3-4) E is a Prime.

(3-5) A = B

(4-1) A is a Prime.

(4-2) B is a Prime.

(4-3) A is a Square.

(4-4) B is a Square.

What is the value of N?

View Solution

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6/18

NUMERICALLOGIC--- META-PUZZLESDesignedby Robert A. Kraus

Backto LOGICPUZZLES

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To solve the puzzles you should have fairly good ability and knowledge of basic logic, high-school algebra, and the

simplest ideas of elementary number theory (e,g, primes, squares, divisibilty, etc.). Most of the problems require the compilatio

and sorting of large lists of numbers and the use of a good spreadsheet program is necessary.The type and difficulty of the

problems varies, some are quite long and difficult.

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InheiritanceEnvelopes

A wealthy man had three sons all of whom were quite good at math and logic. To get a share of his inheiritance each had to

correctly determine a positive integer which he had chosen. He told them that the number had four different non-zero decimal dig

in ascending order.

He prepared three sealed envelopes each of which contained a number. The first contained the product of the four digits, the seco

contained the sum of the squares of the four digits, the third contained the sum of the product of the first two digits and the produ

of the last two digits, and the envelopes were clearly marked as such. He showed the three envelopes to the three sons and had the

each take one at random.

The sons were stationed at three different computers so that they couldn't communicate with one another (but were linked to the

father's computer). After one hour they could submit a number or decline. Anyone who submitted a wrong answer would be

eliminated and get nothing. If one or more submitted the correct answer they would each receive a share of the inheritance, and thcontest would end with the others getting nothing. If no one submitted the correct answer they would be instructed to work on the

problem for another hour. The process would repeat as often as necessary. Each of the sons decided not to submit an answer unle

they sure it was correct.

At the end of the first hour no one had submitted an answer.

At the end of the second hour no one had submitted an answer.

At the end of the third hour no one had submitted an answer.

At the end of the fourth hour all three of them submitted the correct answer!

Can you determine the number?

View Solution

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7/18

Math Class

The math teacher addressed the class. "I have chosen a positive integer (base 10). It has 4 digits, none of which are 0. I have

calculated the sum of its digits, the sum of the squares of its digits, and the product of its digits, and no 2 of these 3 quantities ha

same number of digits. Consider the following 9 statements:

(1) The number is a prime.

(2) The sum of its digits is a prime.(3) The sum of the squares of its digits is a prime.

(4) The number is a square.

(5) The sum of its digits is a square.

(6) The sum of the squares of its digits is a square.

(7) The number is triangular.

(8) The sum of its digits is triangular.

(9) The sum of the squares of its digits is triangular.

You must use them to determine the number."

"But they can't all be true!", interjected one of the students.

"I never said they were! Some of the statements are true and some are false."

"Well we will need more information. Tell us which are true and which are false.""If I told you that you would easily be able to determine the number!"

"Well at least tell us how many are true."

"If I told you that now, you would be able to determine the number too easily!"

"Well, what more can you tell us?"

"Nothing! I have told you enough!"

What was the number?

View Solution

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8/18

LongDivision

A high-school math teacher chose 4 of his best students to participate in an extra-credit project. He told them he had chos

a long division calculation, expressed as N=D*Q+R, in which N, the dividend, was a 4-digit perfect square; D, the divisor, wa

3-digit perfect square; Q, the quotient, is defined as the integral part of N/D, and R, the remainder, is defined as N-D*Q.

Each of the 4 students was given the value of one of the 4 quantities, N,D,Q,R, and was told to calculate the 3 others. Eastudent knew which quantity each of the others was given, but not its value.

The teacher told them to begin working but not to speak until he questioned them.

After 15 minutes he asked if anyone found the answers. Only one student had.

"Your answers are correct", he said, "the other 3 should keep working."


"Your answers are correct", he said, "the other 2 should keep working."


"Your answers are correct", he said, "the remaining student should keep working."

A few minutes later the last student turned in the correct answers.

Afterward, the teacher told a friend about the test and asked if he could determine the 4 numbers. After working on it for a

day the friend complained, "I found several possible solutions. I need more information. Could you tell me which student kne

which quantity?"

The teacher replied, "That's far too much information! In fact if I just told you which quantity the last student knew you co

find the numbers."

What were the 4 numbers?

View Solution

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INDEXEDNUMBERPUZZLES

Designedby Robert A. Kraus

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9/18

IndexedNumberPuzzle #1

Here is a numbered list of statements, some true, some false, which refer to a specific number (positive integer,

base 10). It just so happens that if a statement is true then its index number appears among the number's digits,

and if a statement is false then its index number does not appear among the number's digits. Find the number.

(0) The sum of the number's digits is a prime.

(1) The product of the number's digits is odd.

(2) Each of the number's digits is less than the next digit (if there is one).

(3) No two of the number's digits are equal.

(4) None of the number's digits is greater than 4.

(5) The number has less than 6 digits.

(6) The product of the number's digits is not divisible by 6.

(7) The number is even.

(8) No two of the number's digits differ by 1.

(9) At least one of the number's digits is equal to the sum of 2 other digits. (Any of the digits may be equal, as

long as all 3 digits are distinct.).

The solution is unique.

View Solutiuon

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10/18





(0) The number contains exactly 2 different digit values.

(1) No 2 digits differ by 4, 5, or 6.

(2) More than 1 digit is a square.

(3) Exactly 1 digit value appears more than once.

(4) All digits are even.

(5) The number is a palindrome.

(6) No prime digit divides a different-valued digit.

(7) No digit value appears more than twice.

(8) The number contains all digit values between the largest and smallest.

(9) All adjacent digits differ by 1.


Terminology Example:

The number 77744 has 5 distinct digits taking on 2 different digital values.

The 1st, 2nd, and 3rd digits have a digit value of 7 and the 4th and 5th digits have a digit value of 4.

View Solution

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11/18





A "triple" means any sub-string of 3 consecutive digits, (consecutive by position, not value).

(0) The number does not have 2 equal adjacent digits.

(1) At least one digit is an odd prime.

(2) The number is odd.

(3) The third digit of any triple is equal to the first digit plus the second digit (mod 10).

(4) The last 3 digits are primes.

(5) The third digit of any triple is equal to the first digit minus the second digit (mod 10).

(6) Exactly one digit value is prime.

(7) The third digit of any triple is equal to the second digit minus the first digit (mod 10).

(8) The last 2 digits are both squares.

(9) The last 2 digits are both even.


View Solution

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12/18





Terminology:

"DV" means "digit value", which refers to one of the 10 numbers 0,1,2,3,4,5,6,7,8,9; as opposed to "digit",

which refers to a position of a DV within a number. (2444233 has 7 digits but only 3 DVs. The 3 DVs (2,3,4)

are consecutive DVs, but do not appear as consecutive digits; the 3rd,4th,5th digits are consecutive regardless of

their DVs, (which happen to be 4,4,2 and are not consecutive).

A "triple" means any sub-string of 3 consecutive digits, (consecutive by position, not value).

A "balanced triple" is any triple for which the middle digit is equal to the sum (mod 10) of the other 2 digits.

Otherwise, it is an "unbalanced triple".

(0) Contains 3 of the DVs less than 4.

(1) Does not contain 2 equal digits.

(2) Does not contain 3 digits whose sum is even.

(3) Does not contain an unbalanced triple.

(4) If a DV occurs as a digit, then its square (mod 10) also occurs as a digit.

(5) Contains all of the odd DVs greater than 4.

(6) The last digit equals the square (mod 10) of the first digit.

(7) If a DV occurs as a digit, then 10 minus the DV also occurs as a digit.

(8) Contains 3 consecutive DVs.

(9) Contains at least 6 different DVs.


View Solution

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13/18





Terminology Example: (The abbreviation DV means digital value.)

The number 77744 has 5 distinct digits taking on 2 different digital values.

The 1st, 2nd, and 3rd digits have a DV of 7 and the 4th and 5th digits have a DV of 4.

(0) The product of all of the digits is not less than the 2 raised to the power of the number of digits.

(1) The sum of all of the digits is divisible by the number of digits, which is less than the DV of the first digit.

(2) The number is a palindrome.

(3) The number has at least 3 DV's and the largest DV is equal to the sum of the 2 smallest DV's.

(4) The number contains at least 2 prime DV's.

(5) The last 2 digits have equal DV's.

(6) The largest DV is a prime.

(7) The number is divisible by the DV of each of its digits.(8)The number is divisible by at least 3 different prime numbers.

(9)The number has at least 3 DV's and the smallest DV is equal to the difference between the 2 largest DV's.


View Solution

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Logic Test --- RobertA. Kraus2002

Answer each of the 8 questions with a letter from A to D.

The word "answer" in the test refers to YOUR answer, not some hypothetical "best" answer.

After choosing the 8 answers score the test by comparing each question with your answers.

Score 1 point for each question answered correctly, 0 otherwise.

Keep re-taking the test, trying to get the highest possible score.

(Of course you must know what the highest possible score is in order to correctly score

the last question!)

(1) The next question with the same answer as this one is:

(A) 2 (B) 3 (C) 4 (D) 5

(2) The first question with answer C is:

(A) 1 (B) 2 (C) 3 (D) 4

(3) The last question with answer A is:(A) 5 (B) 6 (C) 7 (D) 8

(4) The number of questions with answer D is:

(A) 1 (B) 2 (C) 3 (D) 4

(5) The answer occuring the most is: (if tied, first alphabetically)

(A) A (B) B (C) C (D) D

(6) The first question with the same answer as the question following it is:

(A) 2 (B) 3 (C) 4 (D) 5

(7) The answer occuring the least is: (if tied, last alphabetically)


14/18

(A) A (B) B (C) C (D) D

(8) The highest possible score on this test is:

(A) 5 (B) 7 (C) 6 (D) 8

View Solution

Back to Logic Puzzles

DIGITALPROBLEMSDesignedby RobertA. Kraus

Backto MATHPUZZLES

BalancedNumbers

A balanced number is a positive integer (base 10) whose digital product is equal to its digital sum. Examples: 123,

2141, 31113, 51121.

Verify the following facts about balanced numbers:

(1) A balanced number cannot have 0 as a digit.

(2) All 1-digit numbers are balanced.

(3) There is only one 2-digit balanced number (find it!)

(4) Any balanced number of 3 or more digits must contain the digit 1 (usually lots of ones!)

(5) Any permutaion of the digits of a balanced number is also balanced.

In view of (5) let's call a set of all balanced numbers which are digital permutaions of one-another a clan, and the

largest member of the clan is its leader. (Digital permutation is clearly an equivalence relation and the clans are its

equivalence classes.) Note that the digits of a clan leader are in monotone decreasing order, i.e., each digit is less

than or equal to the previous one. Examples: {123,132,213,231,312,321} is a clan whose leader is 321;

{1124,1142,1214,1241,1412,1421,2114,2141,2411,4112,4121,4211} is a clan whose leader is 4211.

Also let's call any digit greater than one a big digit. The leading string of big digits of a clan leader is called its

head, and the trailing string of ones (if any) is called its tail.

(6) Any number consisting of only big digits in monotone decreasing order forms the head of exactly one clan

leader.

(7) What do the following two sequences related to clan leaders represent? (Hint: Do n-digit balanced numbersexist for every n?)

(A) 9,1,1,1,3,1,2,2,2,1,2,1,3,1,1,1,3,1,2,1,2,1,2,...

(B) 24,34,35,44,48,66,67,75,76,78,80,82,91,92,97,103,104,105,106,112,...

View Solution
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15/18

NumberCycles

PART 1:

Find a cycle of five 4-digit numbers such that the last 2 digits of each number are equal to the first 2 digits

of the next number in the cycle. Each of the 5 numbers must be exactly one of the following types (with

each of the 5 types being represented exactly once): Square, Cube, Triangular, Prime, Fibonacci. The

solution is unique.

PART 2:

Find a cycle of six 4-digit numbers such that the last 2 digits of each number are equal to the first 2 digits

of the next number in the cycle. Each of the 6 numbers must be exactly one of the following types (with

each of the 6 types being represented exactly once): Square, Cube, Triangular, Prime, Fibonacci, Power-of-

Two. The solution is unique.

View Solution

Pandigital

Puzzler

Find a positive

integer (base

10) which has

the following

fourproperties:

(1) Each of the

10 digits (0 to

9) appears

exactly once in

the number.

(2) For each

pair of digits

whose sum is

9, the number

of other digits

positionedstrictly

between the

pair is equal to

the smaller

digit of the

pair.

(3) The sum of

each pair of
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16/18

digits

positioned at

the same

distance from

opposite ends

of the number

is a primenumber.

(4) The

difference

between any 2

adjacent digits

is greater than

1.

The solution is

unique.

ViewSolution
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17/18

Tale of The MissingLists

An absent-minded math professor was telling his eccentric colleague about losing some of his paperwork.

"I can't find my special number lists," he complained.

"What was so special about them?" replied his colleague.

"Well," he explained, "I call a list of numbers 'special' if it has the following 7 properties:"(1) All of the numbers on the list are positive integers, base 10.

(2) There is more than one number on the list.

(3) All of the numbers on the list have the same number of digits.

(4) No two of the numbers on the list are equal.

(5) The smallest number on the list is equal to the total number of digits of all the numbers on the list.

(6) The largest number on the list is equal to the sum of all of the digits of all of the numbers on the list.

(7) The number of items on the list is equal to the largest number divided by the smallest number.

"There must be thousands of special lists," replied his colleague.

"Hmm, I'm not sure about that, but I was only interested in four of them."

"What four?"

"What for? Well, I divide the sum of the numbers on each list by 100 and the four remainders comprise the

four-number combination to my office safe. The safe is locked and I can't remember the combinationwithout them!"

"No, I meant which four!"

"Well obviously I can't remember the lists, but I do recall their properties. First, all of the numbers on a list

that lie strictly between the smallest and largest numbers I will call the middle-numbers. Now all of the

middle-numbers on the first list are primes, and all of the middle-numbers on the second list are composite.

Furthermore, the two largest middle-numbers on the third list do not appear on the first or second lists.

Finally, the largest middle-number on the fourth list does not appear on any of the other three lists, and the

smallest middle-number of the fourth list is not equal to the smallest middle-number of any of the other three

lists."

"Hmm, let me think about that. By the way, what do you have in your safe?"

"I can't remember! But I usually lock up important papers that I am currently working on."

The next day the eccentric professor approached the absent-minded professor while waving a handful of

papers. "I found your lists!" he shouted."Where in the world did you find them? I've looked everywhere!"

"I found them in your safe!"

Questions:

(1) What is the combination to the safe?

(2) How many possible "special" lists are there?

View Solution
http://www.geocities.com/bob_kraus_2000/SOL_TALEOFTHEMISSINGLISTS.htmlhttp://www.geocities.com/bob_kraus_2000/SOL_TALEOFTHEMISSINGLISTS.html


18/18

A SimpleSet

Find a non-empty set of positive integers, base 10, no two equal, such that:

(1) For any number N in the set, none of the numbers in the set has N digits.

(2) For any number N in the set, the sum of the squares of the digits of N is also in the set.

(3) The number of numbers in the set is also in the set.(4) The largest number in the set is equal to the sum of all of the other numbers in the set.

Among all such sets find one with the fewest numbers, and among those,

find the one with the smallest largest number.

View Solution

Digitally-Deluxe

Make a list of positive integers, base 10, no two equal.

Count the number of times the digit 0 appears anywhere among the numbers of the list.Do the same for the digits 1,2,3,4,5,6,7,8,9. This yields 10 numbers (counts).

The 10 counts must all be greater than zero, with no 2 being equal.

Such a list will be called digitally-deluxe. The sum of all of the numbers on the list

will be called its total.

Part 1 (inclusive list): find a digitally-deluxe list in which each of the 10 counts

appear as numbers on the list. Among all such lists find one with the smallest total.

Part 2 (exclusive list): find a digitally-deluxe list in which none of the 10 counts

appear as numbers on the list. Among all such lists find one with the smallest total.

View Solution
http://www.geocities.com/bob_kraus_2000/SOL_SIMPLESET.htmlhttp://www.geocities.com/bob_kraus_2000/SOL_DIGITALLY-DELUXE.htmlhttp://www.geocities.com/bob_kraus_2000/SOL_SIMPLESET.htmlhttp://www.geocities.com/bob_kraus_2000/SOL_DIGITALLY-DELUXE.html

Numerical Logic Puzzles

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