## Arithmetic

An archive of questions and answers that may be of interest to puzzle enthusiasts.
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Question 1 - arithmetic:
A customer at a 7-11 store selected four items to buy, and was told that the cost was \$7.11. He was curious that the cost was the same as the store name, so he inquired as to how the figure was derived. The clerk said that he had simply multiplied the prices of the four individual items. The customer protested that the four prices should have been ADDED, not MULTIPLIED. The clerk said that that was OK with him, but, the result was still the same: exactly \$7.11.

What were the prices of the four items? Show Answer

Question 2 - arithmetic.progression:
Is there an arithmetic progression of 20 or more primes? Show Answer

Question 3 - day.of.week:
It's restful sitting in Tom's cosy den, talking quietly and sipping a glass of his Madeira.

I was there one Sunday and we had the usual business of his clock. When the radio gave the time at the hour, the Ormolu antique was exactly 3 minutes slow.

"It loses 7 minutes every hour", my old friend told me, as he had done so many times before. "No more and no less, but I've gotten used to it that way."

When I spent a second evening with him later that same month, I remarked on the fact that the clock was dead right by radio time at the hour. It was rather late in the evening, but Tom assured me that his treasure had not been adjusted nor fixed since my last visit.

What day of the week was the second visit?

Question 4 - clock/palindromic:
How many times per day does a digital clock display a palindromic number? Show Answer

Question 5 - clock/reversible:
How many times per day can the hour and minute hands on an analog clock switch roles and still signify a valid time, ignoring the second hand? Show Answer

Question 6 - clock/right.angle:
How many times per day do the hour and minute hands of a clock form a right angle? Show Answer

Question 7 - thirds:
Do the 3 hands on a clock ever divide the face of the clock into 3 equal segments, i.e. 120 degrees between each hand? Show Answer

Question 8 - consecutive.composites:
Are there 10,000 consecutive non-prime numbers? Show Answer

Question 9 - consecutive.product:
Prove that the product of three or more consecutive positive integers cannot be a perfect square. Show Answer

Question 10 - consecutive.sums:
Find all series of consecutive positive integers whose sum is exactly 10,000. Show Answer

Question 11 - conway:
Describe the sequence a(1)=a(2)=1, a(n) = a(a(n-1)) + a(n-a(n-1)) for n>2. Show Answer

Question 12 - digits/6.and.7:
Does every number which is not divisible by 5 have a multiple whose only digits are 6 and 7? Show Answer

Question 13 - digits/all.ones:
Prove that some multiple of any integer ending in 3 contains all 1s. Show Answer

Question 14 - digits/arabian:
What is the Arabian Nights factorial, the number x such that x! has 1001 digits? How about the prime x such that x! has exactly 1001 zeroes on the tail end. (Bonus question, what is the 'rightmost' non-zero digit in x!?) Show Answer

Question 15 - digits/circular:
What 6 digit number, with 6 different digits, when multiplied by all integers up to 6, circulates its digits through all 6 possible positions, as follows:
ABCDEF * 1 = ABCDEF
ABCDEF * 3 = BCDEFA
ABCDEF * 2 = CDEFAB
ABCDEF * 6 = DEFABC
ABCDEF * 4 = EFABCD
ABCDEF * 5 = FABCDE Show Answer

Question 16 - digits/divisible:
Find the least number using 0-9 exactly once that is evenly divisible by each of these digits. Show Answer

Question 17 - digits/equations/123456789:
In how many ways can "." be replaced with "+", "-", or "" (concatenate) in .1.2.3.4.5.6.7.8.9=1 to form a correct equation? Show Answer

Question 18 - digits/equations/1992:
1 = -1+9-9+2. Extend this list to 2 through 100 on the left side of the equals sign. Show Answer

Question 19 - digits/equations/24:
Form an expression that evaluates to 24 that contains two 3's, two 7's, and zero or more of the operators +, -, *, and /, and parentheses. What about two 4's and two 7's, or three 5's and one 1, or two 3's and two 8's? Show Answer

Question 20 - digits/equations/383:
Make 383 out of 1,2,25,50,75,100 using +,-,*,/. Show Answer

Question 21 - digits/equations/find:
Write a program for finding expressions built out of given numbers and using given operators that evaluate to a given value, or listing all possible values. Show Answer

Question 22 - digits/extreme.products:
What are the extremal products of three three-digit numbers using digits 1-9? Show Answer

Question 23 - digits/labels: You have an arbitrary number of model kits (which you assemble for fun and profit). Each kit comes with twenty (20) stickers, two of which are labeled "0", two are labeled "1", ..., two are labeled "9". You decide to stick a serial number on each model you assemble starting with one. What is the first number you cannot stick. You may stockpile unused numbers on already assembled models, but you may not crack open a new model to get at its stickers. You complete assembling the current model before starting the next. Show Answer

Question 24 - digits/least.significant/factorial:
What is the least significant non-zero digit in the decimal expansion of n!? Show Answer

Question 25 - digits/least.significant/tower.of.power:
What are the least significant digits of 9^(8^(7^(6^(5^(4^(3^(2^1))))))) ? Show Answer

Question 27 - digits/most.significant/powers:
What is the probability that 2^N begins with the digits 603245? Show Answer

Question 28 - digits/nine.digits:
Form a number using 0-9 once with its first n digits divisible by n. Show Answer

Question 29 - digits/palindrome:
Does the series formed by adding a number to its reversal always end in a palindrome? Show Answer

Question 30 - digits/palintiples:
Find all numbers that are multiples of their reversals. Show Answer

Question 31 - digits/power.two:
Prove that for any 9-digit number (base 10) there is an integral power of 2 whose first 9 digits are that number. Show Answer

Question 32 - digits/prime/101:
How many primes are in the sequence 101, 10101, 1010101, ...? Show Answer

Question 33 - digits/prime/all.prefix:
What is the longest prime whose every proper prefix is a prime? Show Answer

Question 34 - digits/prime/change.one:
What is the smallest number that cannot be made prime by changing a single digit? Are there infinitely many such numbers? Show Answer

Question 35 - digits/prime/prefix.one:
2 is prime, but 12, 22, ..., 92 are not. Similarly, 5 is prime whereas 15, 25, ..., 95 are not. What is the next prime number which is composite when any digit is prefixed? Show Answer

Question 36 - digits/reverse:
Is there an integer that has its digits reversed after dividing it by 2? Show Answer

Question 37 - digits/rotate:
Find integers where multiplying them by single digits rotates their digits one position, so that the last digit become the first digit. Show Answer

Question 38 - digits/sesqui
Find the least number where moving the first digit to the end multiplies by 1.5. Show Answer

What squares remain squares when their leading digits are incremented? Show Answer

Question 40 - digits/squares/length.22:
Is it possible to form two numbers A and B from 22 digits such that A = B^2? Of course, leading digits must be non-zero. Show Answer

Question 41 - digits/squares/length.9
Is it possible to make a number and its square, using the digits from 1 through 9 exactly once? Show Answer

Question 42 - digits/squares/three.digits
What squares consist entirely of three digits (e.g., 1, 4, and 9)? Show Answer

Question 43 - digits/squares/twin
Let a twin be a number formed by writing the same number twice, for instance, 81708170 or 132132. What is the smallest square twin? Show Answer

Question 44 - digits/sum.of.digits:
Find sod ( sod ( sod (4444 ^ 4444 ) ) ). Show Answer

Question 45 - digits/zeros/million:
How many zeros occur in the numbers from 1 to 1,000,000? Show Answer

Question 46 - digits/zeros/trailing:
How many trailing zeros are in the decimal expansion of n!? Show Answer

Question 47 - magic.squares:
Are there large squares, containing only consecutive integers, all of whose rows, columns and diagonals have the same sum? How about cubes? Show Answer

Question 48 - pell:
Find integer solutions to x^2 - 92y^2 = 1. Show Answer

Question 49 - subset:
Prove that all sets of n integers contain a subset whose sum is divisible by n. Show Answer

Question 50 - sum.of.cubes:
Find two fractions whose cubes total 6. Show Answer

Question 51 - sums.of.powers:
Partition 1,2,3,...,16 into two equal sets, such that the sums of the numbers in each set are equal, as are the sums of their squares and cubes. Show Answer

Question 52 - tests.for.divisibility/eleven:
What is the test to see if a number is divisible by eleven? Show Answer

Question 53 - tests.for.divisibility/nine:
What is the test to see if a number is divisible by nine? Show Answer

Question 54 - tests.for.divisibility/seven:
What is the test to see if a number is divisible by seven? Show Answer

Question 55 - tests.for.divisibility/three
What is the test to see if a number is divisible by three? Show Answer