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Add To One-thousand      Problem ID: 136 (Dec 2003)
How many pairs of positive integers add to make one-thousand?
Add To Six      Problem ID: 29 (Jan 2001)
Using positive integers, how many different sums add to make six?
Ancient Riddle      Problem ID: 2 (Aug 2000)
Can you discover the password to unlock the tests?
Area Of Arrow      Problem ID: 65 (Feb 2002)
Can you find the area of the arrow?
Arithmetic Ring      Problem ID: 117 (May 2003)
Show how to fill a ring with the digits 1 to 8 so that no two adjacent numbers are consecutive.
Arithmetic Volume      Problem ID: 198 (10 Jan 2005)
How many cuboids exist for which the volume is less than 100 units3 and the integer side lengths are in an arithmetic sequence?
As Easy As 1234      Problem ID: 13 (Sep 2000)
Using the digits 1, 2, 3, and 4 (no more and no less), can you make all of the integers from 1 to 25?
Average Problem      Problem ID: 232 (31 Jul 2005)
Can you determine the set of five numbers given the average clues?
Balancing Scales      Problem ID: 91 (Dec 2002)
How many different weights can you weigh using 2 kg, 3 kg and 5 kg and a set of balancing scales?
Beam Of Light      Problem ID: 62 (Jan 2002)
Which corner will the beam of light emerge from the prism?
Bigger Digit      Problem ID: 217 (30 Mar 2005)
For how many 2-digit numbers is the first digit greater than the second digit?
Birds And Bunnies      Problem ID: 350 (17 Apr 2009)
Can you find out how many birds are in the cage?
Birthday Party      Problem ID: 273 (21 Apr 2006)
Can you work out how many people attended the party?
CD To Tape      Problem ID: 18 (Oct 2000)
The most efficient method of arranging songs on a tape.
Children In A Circle      Problem ID: 30 (Jan 2001)
How many children are standing in the circle?
Chocolate Offer      Problem ID: 122 (Oct 2003)
Can you would out how much ten chocolate bars would cost?
Class Distinction      Problem ID: 192 (11 Dec 2004)
Can you discover the minimum number of children in the class with brown hair and are right handed?
Coin Problem      Problem ID: 105 (Mar 2003)
Using the information given, can you work out how many of each coin I have?
Connected Pentagon      Problem ID: 177 (Oct 2004)
How many triangles are in a fully connected pentagon?
Counting Coins      Problem ID: 150 (Feb 2004)
What is the maximum amount of money that you could have in your pocket and not able to make exactly £2?
Counting Nines      Problem ID: 100 (Feb 2003)
How many nines are there in all of the numbers from 1 to 100?
Counting Squares      Problem ID: 107 (Mar 2003)
How many squares are there altogether on a standard chessboard?
Counting Triangles      Problem ID: 42 (Apr 2001)
How many triangles are there in the diagram?
Counting Triangles Again      Problem ID: 69 (Mar 2002)
How many triangles are there in the diagram?
Counting Up To One Thousand      Problem ID: 323 (30 May 2007)
If all the counting numbers from 1 to 1000 were written out how many digits would be written down?
Crossing Trains      Problem ID: 96 (Jan 2003)
Can you discover how many miles from the station did the two trains cross?
Cross Country Race      Problem ID: 182 (01 Nov 2004)
From the information given, can you work out how many schools took part in the cross country race?
Cuddly Wuddly Tribe      Problem ID: 335 (27 Mar 2008)
What question can you ask to ensure he gets to the Cuddly-Wuddly tribe?
Defensive Bishop      Problem ID: 144 (Jan 2004)
What is the least number of Bishop pieces required to protect every square on a 8 by 8 chessboard?
Dice Problem      Problem ID: 68 (Mar 2002)
What is the sum of the visible faces on the dice?
Different Speeds      Problem ID: 225 (04 Jun 2005)
How long will the tortoise take to catch up with the hare?
Different Totals      Problem ID: 85 (Nov 2002)
By ensuring that the row and column in the grid have the same total, how many different totals can this be done with?
Digital Square Sum      Problem ID: 320 (14 Apr 2007)
How many years during the twenty-first century has a square digital sum?
Digit Product      Problem ID: 17 (Oct 2000)
Investigating products of digits.
Digit Sum      Problem ID: 206 (17 Feb 2005)
How many 3-digit numbers exist for which the sum of the digits is six?
Dividing 2 And 3      Problem ID: 45 (May 2001)
How many numbers below one hundred are divisible by both 2 and 3?
Divisible Consecutive Sums      Problem ID: 33 (Feb 2001)
Which numbers that are divisible by 10 can also be written as the sum of four consecutive integers?
Easter Eggs      Problem ID: 111 (Apr 2003)
How many Easter eggs will be bought by the family in total?
Efficient Measures      Problem ID: 110 (Apr 2003)
Which lengths, using one 2 m length and two 3 m lengths can you measure directly?
Even Digits Multiple Of Nine      Problem ID: 327 (05 Jul 2007)
Find the smallest multiple of nine containing only even digits.
False Ceilings      Problem ID: 59 (Dec 2001)
What length of metal strips will be required to tile the roof of a 3x4 room?
Father And Child      Problem ID: 77 (May 2002)
Can you work out how old I am?
Faulty Scales      Problem ID: 221 (24 May 2005)
Can you discover the true weight of boy's sister?
Four Fours      Problem ID: 14 (Sep 2000)
Using four 4's (no more and no less), can you make all of the integers from 1 to 25?
Fraction Of A Square      Problem ID: 46 (May 2001)
What fraction of the square is shaded?
Fraction Product      Problem ID: 90 (Dec 2002)
Can you work out the exact value of the fraction product?
Heavy Baggage      Problem ID: 116 (May 2003)
Can you calculate how much would a passenger carrying 80 kg of luggage would be charged?
Hidden Palindrome      Problem ID: 163 (Apr 2004)
Can you find the palindrome using the clues?
Hollow Cube      Problem ID: 27 (Dec 2000)
How many cubes remain in the 5 × 5 × 5 cube?
Incomplete Fractions      Problem ID: 26 (Dec 2000)
Can you find the missing numerator and denominator to complete the fraction sum?
In The Beginning      Problem ID: 5 (Aug 2000)
See if you can find the seed of this secret message.
Kitchen Floor      Problem ID: 92 (Dec 2002)
How many white tiles will be needed to complete the kitchen floor design?
Lines Of Symmetry      Problem ID: 38 (Mar 2001)
By adding a single unit square can you give the diagram a line of reflective symmetry?
Logically Addressed Question      Problem ID: 258 (09 Jan 2006)
What question can be used to logically deduce which address belongs to which parent?
Making Primes      Problem ID: 60 (Jan 2002)
Can you make 2 prime numbers using the digits 1, 2, 3 and 5?
Meeting Trains      Problem ID: 261 (29 Jan 2006)
Can you work out how far apart the trains were twenty minutes before passing each other?
Mowing The Lawn      Problem ID: 64 (Feb 2002)
How long will it take for two people to mow the lawn?
Number Chain      Problem ID: 80 (Oct 2002)
Can you discover how the number chain works?
Odd Product      Problem ID: 61 (Jan 2002)
How many 2-digit numbers have an odd product?
On Target      Problem ID: 78 (May 2002)
Can you work out who hit the centre target?
Overlapping Rectangles      Problem ID: 289 (22 Sep 2006)
Can you work out which shape has the greatest perimeter?
Painted Faces      Problem ID: 129 (Nov 2003)
On a 3x3x3 cube, how many cubes have exactly two faces painted?
Palindromic Distance      Problem ID: 267 (11 Feb 2006)
Can you work out how fast Julie was travelling?
Palindromic Years      Problem ID: 143 (Jan 2004)
What is the largest gap between two consecutive palindromic years?
Pandigital Minimum Difference      Problem ID: 202 (24 Jan 2005)
What is the minimum difference between two 5-digit pandigitals?
Pathed Pathways      Problem ID: 57 (Dec 2001)
How many different ways can a 5x2 pathway be pathed with 2x1 paving stones?
Peaceful Queens      Problem ID: 50 (Oct 2001)
How many peaceful queens can you place on a 4x4 chessboard?
Pentominoes      Problem ID: 15 (Sep 2000)
A geometrical investigation using pentominoes.
Perimeter Sequence      Problem ID: 76 (May 2002)
What is the perimeter of the tenth pattern in the sequence?
Picture Frame      Problem ID: 308 (04 Feb 2007)
Can you work out the minimum length of material required to make a frame for a photograph?
Pile Of Oranges      Problem ID: 34 (Feb 2001)
What is the biggest square based pyramid you can make from 1000 oranges?
Postage Stamps      Problem ID: 58 (Dec 2001)
How many ways can a £1 stamp book be filled with 9p first class and 7p second class stamps?
Product Of Ones      Problem ID: 23 (Nov 2000)
Investigating products that consist entirely of ones.
Product Of Zero      Problem ID: 37 (Mar 2001)
If you continue multiplying the digits in any 2-digit number, which starting numbers will finish on zero?
Quarter Square      Problem ID: 353 (23 Jul 2009)
How many ways can exactly two quarter squares be shaded?
Rectangular Arrangements      Problem ID: 49 (Oct 2001)
How many ways can you place a 3x2 block on a 4x3 grid?
Reverse Difference      Problem ID: 277 (13 May 2006)
Find the missing digit in the subtraction.
Reverse Prime      Problem ID: 156 (Mar 2004)
How many two digit primes can you find for which their reverse is also prime?
Right Angle Reasoning      Problem ID: 252 (12 Dec 2005)
Explain why the hypotenuse is the longest side in a right angle triangle
Rotational Years      Problem ID: 41 (Apr 2001)
How many years since the birth of Christ read the same upside-down?
Seeing Clearly      Problem ID: 8 (Aug 2000)
Can you discover the meaning of the secret message written in the book?
Shaded Area      Problem ID: 22 (Nov 2000)
What area of the squares are shaded?
Shaded Grid      Problem ID: 86 (Nov 2002)
How many different ways can a 2x2 grid be shaded?
Shaded Hexagon      Problem ID: 149 (Feb 2004)
By overlapping two equilateral triangles, find the area of the hexagaon
Shaded Rectangle      Problem ID: 106 (Mar 2003)
What fraction of the rectangle is shaded?
Shaded Square      Problem ID: 228 (10 Jul 2005)
What fraction of the square is shaded?
Shading Pattern      Problem ID: 101 (Feb 2003)
By continuing the pattern, can you work out what fraction of a 20x20 grid will be shaded?
Simple Fractions      Problem ID: 72 (Apr 2002)
How many fractions with a denominator equal to 24 cannot be cancelled down?
Skeleton Towers      Problem ID: 53 (Nov 2001)
Find the number of blocks needed to construct the 100th tower.
Square Age      Problem ID: 31 (Jan 2001)
Can you work out when Augustus de Morgan was born?
Strawberry Milk      Problem ID: 10 (Aug 2000)
Can you uncover the identity of the cult leader?
String Fractions      Problem ID: 210 (06 Mar 2005)
Without the use of any other measuring tools, how would you use a 2/3 m length of string to measure 1/2 m?
Sum Digital Sum And Product      Problem ID: 171 (May 2004)
Find all 2-digit numbers that are equal to the sum of its digits product and sum
Sum Of Three      Problem ID: 25 (Dec 2000)
Find three different integers that add to sixteen, where the two smallest add to make the biggest.
System Upgrade      Problem ID: 305 (20 Jan 2007)
Can you work out how much time will be saved by upgrading the computer system?
Tangrams      Problem ID: 16 (Sep 2000)
Creating diagrams using a traditional Chinese tangram.
Terraced Houses      Problem ID: 170 (May 2004)
How many matchsticks will be left if one-thousand houses are made?
Test Average      Problem ID: 197 (21 Dec 2004)
Can you determine the lowest possible percentage score that the student scored in any one of the tests?
The Age Of Her Life      Problem ID: 264 (05 Feb 2006)
From the information, can you discover how old Sarah was when she retired?
To Catch A Liar      Problem ID: 255 (01 Jan 2006)
Can you work out how many of the children are liars?
Triangle Dissection      Problem ID: 19 (Oct 2000)
Can you split an equilateral triangle into six smaller triangles?
Triangular Arithmetic      Problem ID: 95 (Jan 2003)
Can you discover the missing values in the triangular arithmagon?
Triominoes      Problem ID: 21 (Nov 2000)
Arranging triomino pieces of a 4×4 board.
Two-digit Sum And Product      Problem ID: 118 (May 2003)
Investigating the properties of the sum and product of digits.
Weighing Scales      Problem ID: 81 (Oct 2002)
Can you work out the combined weight of the boy, girl, and the dog?
3x3 Magic Square      Problem ID: 366 (15 Nov 2009)
By considering rotations and reflections to be equivalent, prove that there exists only one 3x3 magic square.
Adding Digits      Problem ID: 109 (Mar 2003)
If every combination of the digits 1,2,3,4 was written down, what would be the sum of the numbers formed?
Alternate Fibonacci Ratio      Problem ID: 310 (15 Feb 2007)
What does F$n$+2/F$n$ tend towards as $n$ increases?
Alternating Squares      Problem ID: 119 (May 2003)
Can you work out what fraction of the diagram is shaded?
Annulus      Problem ID: 159 (Mar 2004)
Find the width of the annulus
Area Of Annulus      Problem ID: 341 (26 Jun 2008)
Find the area of the annulus.
Average Matches      Problem ID: 303 (12 Jan 2007)
Is the claim about the average contents of matches in a matchbox correct?
A Number And Its Reciprocal      Problem ID: 297 (17 Dec 2006)
Prove that $x$ + 1/$x$ ≥ 2 for non-negative values of $x$.
Blonde Hair Brown Eyes      Problem ID: 145 (Jan 2004)
What fraction of the class in total have brown eyes?
Box World      Problem ID: 55 (Nov 2001)
How many unique four unit cube arrangements are there?
Calendar Cubes      Problem ID: 103 (Feb 2003)
Show how you would label two wooden cubes to display any date of the month.
Candelabra      Problem ID: 183 (01 Nov 2004)
Find the probability that the three white candles will be adjacent
Chequered Floor      Problem ID: 87 (Nov 2002)
Can you discover the dimensions of the room?
Christmas Trees      Problem ID: 52 (Oct 2001)
Can you work out the most efficient way to plant trees?
Circular Pipes      Problem ID: 36 (Feb 2001)
Using a metre stick how would you find the internal diameter of a large circular pipe?
Circumscribed Triangle      Problem ID: 108 (Mar 2003)
Can you find the area of the circumscribed triangle?
Climbing Stairs      Problem ID: 74 (Apr 2002)
How many ways can you climb ten steps?
Coloured Dice      Problem ID: 124 (Oct 2003)
How should the 2nd die be coloured so that there is an equal chance of getting two faces of the same colour?
Consecutive Cube      Problem ID: 83 (Oct 2002)
Can you prove that the product of three consecutive integers, plus their mean, is always cube?
Consecutive Prime Sum      Problem ID: 332 (19 Nov 2007)
Show that two consecutive primes cannot have a sum that is double a prime.
Counting Sequence      Problem ID: 207 (17 Feb 2005)
Can you determine the 1000th term of the sequence?
Curvature Of The Earth      Problem ID: 44 (Apr 2001)
How far can you see from the top of the Eiffel tower?
Disco Ratios      Problem ID: 131 (Nov 2003)
Can you work out how many girls are at the disco?
Divided Square      Problem ID: 339 (18 Jun 2008)
Prove that the only number of non-overlapping squares you cannot split a unit square into are 2, 3, or 5 smaller squares.
Dodecagon Edges      Problem ID: 73 (Apr 2002)
How many edges does a dodecagon have?
Dotty Squares      Problem ID: 35 (Feb 2001)
How many squares can you draw on a grid measuring 4 dots by 4 dots?
Double An Odd Sum      Problem ID: 283 (23 Jul 2006)
If $S_n$ represents the sum of the first $n$ odd numbers, prove that $4S_n = S_{2n}$.
Egyptian Divisibility      Problem ID: 112 (Apr 2003)
What is the least number which has no remainder when divided by any number from 1 to 10?
Equal Angles      Problem ID: 196 (21 Dec 2004)
Prove that the two angles in the square are congruent.
Expressing Divisibility      Problem ID: 356 (26 Aug 2009)
Prove that $n$($n$ + 1)(2$n$ + 1) is divisible by six for all integer values, $n$.
Factorial Symmetry      Problem ID: 214 (09 Mar 2005)
Solve the equation a!b! = a! + b!
Favourite Mathematician      Problem ID: 4 (Aug 2000)
Can you work out the teacher's favourite mathematician?
Finishing On 150      Problem ID: 39 (Mar 2001)
How many ways can a darts player finish from 150 points?
Five-digit Divisibility      Problem ID: 199 (10 Jan 2005)
Using the digits 1, 2, 3, 4, and 5 to form 5-digit numbers, how many are divisible by 12?
Four Hats      Problem ID: 337 (16 May 2008)
Which of the boys, 1 to 3, is most likely to guess the colour of his hat?
Fractional Steps      Problem ID: 349 (30 Nov 2008)
Can you determine the best way to construct a set of steps leading up to a platform?
Glass In The Door      Problem ID: 165 (Apr 2004)
Can you determine the weight of the glass in the door?
Half Fractions      Problem ID: 151 (Feb 2004)
By concatenating all of the digits 1, 2, 3, 4, and 5 to form the ratio of two numbers, how many ways can you make one-half?
Hexagon Perimeter      Problem ID: 278 (13 May 2006)
Find the perimeter of the hexagon.
Hockey Lockers      Problem ID: 43 (Apr 2001)
From the clues can you work out how many lockers there are?
Ice Cream Cone      Problem ID: 71 (Mar 2002)
Can you find the height of the cone?
Inscribed Circle In Isosceles Triangle      Problem ID: 375 (16 Aug 2010)
Find the radius of the circle inscribed inside the isosceles triangle.
Inscribed Circle In Right Angled Triangle      Problem ID: 364 (03 Nov 2009)
Find the radius of the circle inscribed inside the right angled triangle.
Inscribed Square      Problem ID: 368 (30 Nov 2009)
Find the side length of the square inscribed inside the right angled triangle.
Integer Fraction Product      Problem ID: 114 (Apr 2003)
Can you show when the product of fractions is an integer?
Largest Root      Problem ID: 345 (21 Sep 2008)
Which is greater in value, the square root of two or the cube root of three?
Leaning Ladders      Problem ID: 113 (Apr 2003)
How far above the ground do the two ladders cross?
Letter Product      Problem ID: 88 (Nov 2002)
Can you find the exact value of the letter product?
Lucky Guess      Problem ID: 138 (Dec 2003)
Find the probability of winning the card game
Maximum Square      Problem ID: 152 (Feb 2004)
What are the dimensions of the shaded square inside the 3-4-5 right angle triangle?
Meaning Of Life      Problem ID: 6 (Aug 2000)
What is the password to unlock all secrets?
Measuring Mountains      Problem ID: 132 (Nov 2003)
Use the given information to find the height of the mountain.
Missing Weight      Problem ID: 66 (Feb 2002)
Can you work out the missing weight?
Modest Age      Problem ID: 130 (Nov 2003)
Can you work out how old the teacher really is?
Mystic Rose      Problem ID: 93 (Dec 2002)
How many lines are required to construct the Mystic Rose?
Net Perimeter      Problem ID: 360 (11 Oct 2009)
By considering all the nets of a unit cube, which net has the greatest perimeter?
Never Divides By 5      Problem ID: 318 (07 Apr 2007)
Prove that $x$2 + $x$ + 1 will never divide by 5.
Numbered Discs      Problem ID: 157 (Mar 2004)
Can you work out which numbers are on the two discs?
Numbered Discs 2      Problem ID: 164 (Apr 2004)
If you had discs numbered 1 to 10, how would you separate the discs into the two bags such that no bag contains its double?
Painted Cubes      Problem ID: 51 (Oct 2001)
How many ways can paint a cube with two colours?
Password Cracker      Problem ID: 7 (Aug 2000)
Can you crack the hacker website password?
Pen Problem      Problem ID: 99 (Jan 2003)
Can you work out how many pens the girl bought and how much she paid for each one?
Perfect Cone      Problem ID: 203 (24 Jan 2005)
Can you find the height of the cone?
Perfect Ruler      Problem ID: 70 (Mar 2002)
What is the minimum number of marks required to measure the lengths 1 to 12?
Permuted Sums      Problem ID: 211 (06 Mar 2005)
Find the sum of all 4-digit combinations taken from {1,2,3,4,5}
Prime One Less Than Square      Problem ID: 281 (15 Jul 2006)
Finding primes that are one less than a square.
Prime Square Differences      Problem ID: 94 (Dec 2002)
Can you prove that all primes greater than 2 can be written as the difference of two squares?
Prime Square Divisibility      Problem ID: 67 (Feb 2002)
Can you prove that the square of all primes minus 1 are divisible by 24?
Prime Square Sums      Problem ID: 123 (Oct 2003)
How many primes less than 100 can be written as the sum of two square numbers?
Prime Uniqueness      Problem ID: 48 (May 2001)
Prove that seven is the only prime number that is one less than a perfect cube.
Proportional Difference      Problem ID: 314 (18 Mar 2007)
Show that $a/b = c/d = (a - c)/(b - d)$.
Proportion Of Ones      Problem ID: 189 (28 Nov 2004)
What proportion of 3-digit numbers contain the digit one?
Quadrant Product Divisibility      Problem ID: 222 (24 May 2005)
How would you arrange the numbers 1 to 16 in the grid, such that the product of the numbers in each quadrant is divisible by 16?
Random Routes      Problem ID: 172 (May 2004)
How many different routes can you find through a 4 × 4 grid?
Rectangle Construction      Problem ID: 376 (17 Oct 2010)
Find the connection between the constructed length and the original rectangle.
Rectangular Circles      Problem ID: 253 (12 Dec 2005)
Given the three concentric circles generated by the rectangle show that the area of the inner circle equals the ring generated by the outer circles
Remainder Of One      Problem ID: 233 (31 Jul 2005)
Find the smallest number, greater than 1, which has a remainder of 1 when divided by any of 2, 3, 4, 5, 6, or 7
Reverse Digits      Problem ID: 82 (Oct 2002)
Investigating the divisibility of adding a 2-digit number to its reverse.
Right Time      Problem ID: 137 (Dec 2003)
How many times between 9 a.m. and 3 p.m. is the angle between the hour and minute hand 90o?
Right Triangle Equal Angles      Problem ID: 248 (27 Nov 2005)
In the given right angle triangle prove that the two marked angles are the same size
Rounded Roots      Problem ID: 47 (May 2001)
Investigate the special number machine that square roots and rounds off answers.
Rounding Error      Problem ID: 343 (09 Jul 2008)
Finding the chance of making a rounding error.
Rounding Machine      Problem ID: 97 (Jan 2003)
Can you discover the connection between the input values produce the same output from two different machines?
Running Requirements      Problem ID: 268 (11 Feb 2006)
What should the length of the straight section be on a running track be to meet IAAF requirements?
Same Digits      Problem ID: 229 (10 Jul 2005)
How many 3-digit numbers have two digits the same?
Same Digit Prime      Problem ID: 256 (01 Jan 2006)
Explain why a number made up of the same digit can only be prime if the digit is one AND the number of digits is itself prime
Sand Glass      Problem ID: 226 (04 Jun 2005)
Using one sand glass that measures 9 minutes and another that measures 13 minutes, how would you measure 30 minutes?
Save Rate      Problem ID: 102 (Feb 2003)
From the information given, how can the goal keeper achieve a 50% save rate?
Semi-circle Lunes      Problem ID: 173 (May 2004)
Find the area of the lunes on the semi-circle
Shaded Cross      Problem ID: 125 (Oct 2003)
Find the area of the shaded cross.
Shaded Triangle      Problem ID: 98 (Jan 2003)
Can you find the area of the shaded square inside the triangle?
Simple Fractions Symmetry      Problem ID: 166 (Apr 2004)
For a given denominator, prove that there are always an even number of simple fractions
Slide Height      Problem ID: 294 (26 Nov 2006)
Find the vertical height of the slide
Sloping Square      Problem ID: 179 (Oct 2004)
Find the area of the sloping square.
Snapped Pole      Problem ID: 362 (28 Oct 2009)
Can you determine the point where the pole snapped?
Solid Encryption      Problem ID: 9 (Aug 2000)
Can you decrypt the punchline?
Square And Round Plugs      Problem ID: 370 (24 Dec 2009)
Which fits better... a round plug in a square hole or a square plug in a round hole?
Square Laminas      Problem ID: 56 (Nov 2001)
How many squares can you make from 240 unit tiles?
Square Product      Problem ID: 178 (Oct 2004)
Given that [n(n+1)(n+2)]2 = 3039162537*6, find the value of *
Square Rods      Problem ID: 188 (28 Nov 2004)
Using rods of length 1, 2, 3, ... , n, can you construct a square?
String Of Ones      Problem ID: 287 (09 Sep 2006)
Find the smallest repunit that is divisible by 63.
Taming The Sum      Problem ID: 140 (Dec 2003)
Can you make the sum equal to 100?
Tangential Distances      Problem ID: 351 (17 Apr 2009)
Prove that the tangential distances PS and PT are always equal.
Three Squares      Problem ID: 317 (07 Apr 2007)
Prove that the sum of the given angles are 360 degrees.
Tiled Floor      Problem ID: 54 (Nov 2001)
Can you work out how many tiles fill the room?
Triangle Area      Problem ID: 239 (10 Aug 2005)
Given two points, X and Y, find the area of triangle OXY
Tri Angles      Problem ID: 243 (19 Oct 2005)
Prove that exterior angle of a triangle is equal to the sum of two opposite interior angles
Tunnel Train      Problem ID: 104 (Feb 2003)
How long will it take for the train to completely pass through the tunnel?
Unsorted Socks      Problem ID: 358 (27 Sep 2009)
Find the minimum number of socks which must be taken from the drawers to be certain of finding five matching pairs.
Up Down Left Right      Problem ID: 11 (Aug 2000)
Can you discover the secret of this misleading system?
Weighty Logic      Problem ID: 218 (30 Mar 2005)
Can you use a combination of logic and algebra to determine Goldilock's weight?
X Hits The Spot      Problem ID: 12 (Aug 2000)
Can you discover what X represents?
Absolute Sum      Problem ID: 292 (03 Oct 2006)
How many solutions does the equation $|x| + 2|y| = 100$ have?
Alley Ladders      Problem ID: 127 (Oct 2003)
Can you prove the amazing relationship between the heights of the ladders?
Alternating Sign Square Sum      Problem ID: 262 (29 Jan 2006)
Show that the alternating sign sum of squares produces triangle numbers.
Anonymous Author      Problem ID: 3 (Aug 2000)
Is there a secret hidden in the poetry?
A Radical Proof      Problem ID: 284 (23 Jul 2006)
Use the conjecture based on radicals to prove the Last Theorem of Fermat for $n \ge 6$.
Bag Of Balls      Problem ID: 139 (Dec 2003)
Can you discover the radius of the ball at the bottom of the bag?
Bouncing Ball      Problem ID: 363 (28 Oct 2009)
Where must the ball strike inside the square to return its point of origin?
Cables      Problem ID: 28 (Dec 2000)
What would be the diameter of a circular duct to feed 2, 3 or 4 circular cables?
Coloured Strings      Problem ID: 259 (09 Jan 2006)
Show that it is impossible not to form a triangle with all the edges the same colour
Common Chord      Problem ID: 312 (15 Feb 2007)
Prove that the segment joining the centres, $AB$, is a perpendicular bisector of the common chord $XY$.
Concurrent Circles In A Triangle      Problem ID: 321 (14 Apr 2007)
Prove that circles determined by points on each side of a triangle and each vertex are concurrent.
Consecutive Composites      Problem ID: 325 (26 Jun 2007)
Prove that there exists a sequence of $n$ consecutive composite numbers.
Consecutive Product Square      Problem ID: 184 (01 Nov 2004)
Prove that the product of four consecutive integers is always one less than a perfect square
Converging Root      Problem ID: 160 (Mar 2004)
What form must the constant take in the iterative formula for the limit to be integer?
Corner Circle      Problem ID: 84 (Oct 2002)
Can you find the radius of the circle in the corner?
Cubes And Multiples Of 7      Problem ID: 24 (Nov 2000)
Prove that for any number that is not a multiple of seven, then its cube will be one more or one less than a multiple of 7.
Cuboid Perimeters To Volume      Problem ID: 357 (26 Aug 2009)
Given the three perimeters of a cuboid can you determine its volume?
Divisible By 11      Problem ID: 208 (17 Feb 2005)
Prove that 10n+1 is divisible by 11 iff n is odd
Divisible By 99      Problem ID: 359 (27 Sep 2009)
Find the smallest number made up of the digits 1 through 9 which is divisible by 99.
Double Square Sum      Problem ID: 333 (19 Nov 2007)
Prove that there are infinitely many primitive solutions to the equation $x$2 + $y$2 = 2$z$2.
Equable Rectangles      Problem ID: 340 (18 Jun 2008)
How many rectangles with integral length sides have an area equal in value to the perimeter?
Equal Chance      Problem ID: 146 (Jan 2004)
With an equal chance of picking two discs the same colour, how many discs are in the bag?
Equal Colours      Problem ID: 193 (11 Dec 2004)
Find the number of black discs in the game of chance
Factorial And Power Of 2      Problem ID: 215 (09 Mar 2005)
Prove that a!b! = a! + b! + 2c has a unique solution
Factorial Divisibility      Problem ID: 40 (Mar 2001)
Can you prove that (2n)! is divisible by 22n − 1?
Falling Sound      Problem ID: 346 (21 Sep 2008)
Can you determine the depth of the well by timing how long it takes to hear the splash?
Fibonacci Ratio      Problem ID: 311 (15 Feb 2007)
Prove that the ratio of adjacent terms in the Fibonacci sequence F$n$+1/F$n$ tends towards φ.
Fibonacci Series      Problem ID: 299 (17 Dec 2006)
Find the sum of a modified infinite Fibonacci series.
Fishy Problem      Problem ID: 324 (30 May 2007)
Prove that P(X = L) = P(X = L−1) for all positive integer values of L.
Fraction Reciprocal Sum      Problem ID: 153 (Feb 2004)
Prove that the sum of a proper fraction and its reciprocal can never be integer
Highest Roll Wins      Problem ID: 158 (Mar 2004)
What is the probability of winning the game of chance?
Hops And Slides But Never Square      Problem ID: 372 (07 Aug 2010)
Prove that the minimum number of moves to completely reverse the positions of the coloured counters can never be square.
Imperfect Square Sum      Problem ID: 133 (Nov 2003)
Prove that n4 + 3n2 + 2 is never square.
Impossible Solution      Problem ID: 190 (28 Nov 2004)
Find the conditions for when √ab = √c has a solution.
Increasing Digits      Problem ID: 180 (Oct 2004)
Given a number has strictly increasing digits, what is the probability that it contains 5-digits?
Inscribed Rectangle      Problem ID: 63 (Jan 2002)
Can you find the dimensions of an inscribed rectangle?
Integral Area      Problem ID: 223 (24 May 2005)
For which diagonal lengths is the area of the rectangle, which contains a unit square, integer?
Inverted Logarithm      Problem ID: 315 (18 Mar 2007)
Prove that log$a$($x$)log$b$($y$) = log$b$($x$)log$a$($y$).
Isosceles Trapezium      Problem ID: 79 (May 2002)
Can you find the missing length of the trapzium?
Lunes      Problem ID: 20 (Oct 2000)
Can you find the shaded area of the lunes?
Mean Claim      Problem ID: 306 (20 Jan 2007)
Is the claim about the average number of matches statistically significant?
Mean Proof      Problem ID: 230 (10 Jul 2005)
Prove that (a + b)/2 ≥ √(ab)
Mean Sequence      Problem ID: 212 (06 Mar 2005)
Investigate the nature of the second order recurrence relation.
Multiple Of Six Difference      Problem ID: 330 (13 Jul 2007)
Prove that the difference between the expressions $x$3 + $y$ and $x$ + $y$3 is a multiple of six.
Multiplicatively Perfect      Problem ID: 342 (26 Jun 2008)
Determine the nature of all multiplicatively perfect numbers.
Multiplying Magic Square      Problem ID: 374 (16 Aug 2010)
Show how the values 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be placed in a 3x3 square grid so that the product of each row, column, and diagonal gives the same value.
Odd Power Divisibility      Problem ID: 279 (13 May 2006)
Prove that $6^n + 8^n$ is divisible by 7 iff $n$ is odd.
Odd Vertices      Problem ID: 254 (12 Dec 2005)
Prove that in any graph there will always be an even number of odd vertices
Pandigital Primes      Problem ID: 185 (01 Nov 2004)
How many differen ways can you form five primes using the digits 0 to 9?
Parallelogram Property      Problem ID: 245 (28 Oct 2005)
Prove that a quadrilateral has opposite sides of equal length if and only if it is a parallelogram
Pathway Arrangements      Problem ID: 270 (16 Feb 2006)
Investigating the number of paving a pathway.
Peculiar Perimeter      Problem ID: 89 (Nov 2002)
How many rectangles exist for which the number of tiles on the perimeter are equal to the number of tiles on the inside?
Pentagon Star      Problem ID: 271 (16 Feb 2006)
Can you determine the fraction of the regular pentagon that the star occupies?
Perfect Triangles      Problem ID: 329 (13 Jul 2007)
Prove that all even perfect numbers are triangle numbers.
Perpendicular Construction      Problem ID: 134 (Nov 2003)
Prove that the constructed triangle inside the unit square is isosceles but not equilateral
Platonic Solids      Problem ID: 246 (28 Oct 2005)
Prove that there can be no more than five regular (convex) polyhedra
Polynomial Roots      Problem ID: 373 (07 Aug 2010)
Prove that the roots of the polynomial, $x$n + $c$n-1$x$n-1 + ... + $c$2$x$2 + $c$1$x$ + $c$0 = 0, are irrational or integer.
Powerful Divisibility      Problem ID: 32 (Jan 2001)
Can you prove that the expression 21n − 5n + 8n is divisible by 24?
Powerful Divisor      Problem ID: 319 (07 Apr 2007)
Find the least value of $x$ for which xx + 1 is divisible by 2n.
Power Divisibility      Problem ID: 204 (24 Jan 2005)
Prove that 8n−1 is always divisible by 7
Primes And Square Sums      Problem ID: 120 (May 2003)
Prove that there exists no prime which is one less than a multiple of four that can be written as the sum of two squares.
Prime Difference      Problem ID: 322 (14 Apr 2007)
Given that $a$2 + $b$ and $a$ + $b$2 differ by prime amount find $a$ and $b$.
Prime Exponent And Fourth Power Sum      Problem ID: 269 (11 Feb 2006)
Given that $p$ is prime, when is $4^p + p^4$ prime?
Prime Exponent And Square Sum      Problem ID: 265 (05 Feb 2006)
Given that $p$ is prime, when is $2^p + p^2$ prime?
Prime Form      Problem ID: 285 (01 Aug 2006)
Prove that if $a^n - 1$ is prime and $n \ge 2$ then $a = 2$ and $n$ must be prime.
Prime Power      Problem ID: 290 (22 Sep 2006)
Prove that there only ever exists one prime value $p$ for which $p + 1$ is a perfect power of $k$ and determine the condition for this perfect power to exist.
Prime Power Sum      Problem ID: 191 (28 Nov 2004)
Prove that 2p + 3p can never be a perfect square.
Pythagorean Triplet Product      Problem ID: 301 (02 Jan 2007)
Prove that the for every primitive Pythagorean triplet ($a$, $b$, $c$) that $abc$ is divisible by sixty.
Quadrilateral Parallelogram      Problem ID: 250 (02 Dec 2005)
Show that the midpoints of any quadrilateral form a parallelogram
Radical Axis      Problem ID: 194 (21 Dec 2004)
Prove that the radical axis of two intersecting circles is their common secant.
Rational Quadratic      Problem ID: 247 (28 Oct 2005)
When does the quadratic, ax2pxp = 0, have rational roots?
Rational Roots Quadratic      Problem ID: 274 (21 Apr 2006)
Prove the a quadratic with non-zero integral coefficients can be found for which every arrangement of coefficients yields rational roots.
Reciprocal Symmetry      Problem ID: 241 (16 Oct 2005)
Solve the Diophantine equation c = a/b + b/a
Relatively Prime Permutations      Problem ID: 240 (10 Aug 2005)
Prove that p and p2 cannot be permutations of the value of their Totient function
Reverse Equivalence      Problem ID: 126 (Oct 2003)
Prove that the sum of two 2-digit numbers, for which the reverse of their digits forms a different pair of 2-digit numbers with the same sum, is always divisible by 11.
Sequence Divisibility      Problem ID: 75 (Apr 2002)
Can you show that every term in the sequence is divisibly by 18?
Shaded Octagon      Problem ID: 298 (17 Dec 2006)
Find the fraction of the square which the shaded octagon fills.
Sixteen Circles      Problem ID: 186 (01 Nov 2004)
Find the radius of the shaded circle.
Spider Fly Distance      Problem ID: 201 (10 Jan 2005)
Find the shortest distance from one corner of a cuboid to the opposite corner.
Square Search      Problem ID: 231 (10 Jul 2005)
When is 8p+1 square?
Sum Of Squares And Multiple Of Product      Problem ID: 300 (02 Jan 2007)
Prove that sum of the squares of $x$ and $y$ can never be a multiple of their product.
Sum Product Numbers      Problem ID: 200 (10 Jan 2005)
Prove that N is a sum-product number iff it is composite.
The Fibonacci Series      Problem ID: 352 (17 Apr 2009)
Prove that the sum of the first $n$ terms in the Fibonacci sequence is given by Fn+2 − 1.
Three Circles      Problem ID: 175 (May 2004)
Prove that three touching circles with a common tangent hold a special relationship between their radii
Tick Tock Triangle      Problem ID: 167 (Apr 2004)
Find the probability that the triangle formed by the hands on a clock is isosceles
Trailing Zeroes      Problem ID: 174 (May 2004)
How many trailing zeroes does 1000! contain?
Triangle In Square      Problem ID: 147 (Jan 2004)
Find the maximum area of the triangle region in the unit square
Triangle Median      Problem ID: 213 (06 Mar 2005)
Prove that each median in a triangle is split in the ratio 2:1.
Triangle Reciprocals      Problem ID: 236 (02 Aug 2005)
Find the series of the reciprocals of triangle numbers
Triangle Search      Problem ID: 234 (31 Jul 2005)
When is 8p+1 a triangle number?
Two Squares      Problem ID: 338 (16 May 2008)
Given that $n$ is a positive integer and 2$n$ + 1 and 3$n$ + 1 are perfect squares, prove that $n$ is divisible by 40.
Unexpected Sum      Problem ID: 275 (21 Apr 2006)
Evaluate the exact value of the series.
Uninvited Guest      Problem ID: 354 (23 Jul 2009)
Given that $x$2 + $x$ + 1 = 0, find the value of $x$3.
Unique Circle Equal Angles      Problem ID: 249 (27 Nov 2005)
Show that a unique circle must pass through the vertices of the two triangles
XOR Challenge      Problem ID: 1 (Aug 2000)
Are you able to complete the email challenge?
Zip Codes      Problem ID: 154 (Feb 2004)
How many 5-digit ZIP codes are detour-prone?
15 Degree Triangle      Problem ID: 361 (11 Oct 2009)
Prove that the two smaller angles in the triangle are exactly 15o and 75o respectively.
Algebraic Cosine      Problem ID: 369 (30 Nov 2009)
Prove that cos($x$) is algebraic if $x$ is a rational multiple of Pi.
Almost Equilateral Triangles      Problem ID: 219 (30 Mar 2005)
Prove that infintitely many almost equilateral triangles exist
Composite Fibonacci Terms      Problem ID: 365 (03 Nov 2009)
Prove that for all composite values of $n$ > 4, Fn is composite.
Concurrent Congruent Segments      Problem ID: 309 (04 Feb 2007)
For the given construction prove that the segments are congruent and concurrent with one another.
Concurrent Segments In A Triangle      Problem ID: 316 (18 Mar 2007)
Prove that the product of given ratios in any triangle is always one.
Contains The Origin      Problem ID: 237 (02 Aug 2005)
Which shape is more likely to contain the origin?
Continued Fraction Irreducible Convergents      Problem ID: 286 (01 Aug 2006)
Prove that the convergents generated by the recurrence relation for continued fractions are irreducible.
Continued Fraction Recurrence Relation      Problem ID: 282 (15 Jul 2006)
Prove that the recurrence relation holds for continued fractions.
Counting Digits      Problem ID: 128 (Oct 2003)
How many digits does the number 21000 contain?
Diophantine Challenge      Problem ID: 227 (04 Jun 2005)
Solve the Diophantine equation, x2 + (bx)y = ±1
Euler Rules      Problem ID: 242 (16 Oct 2005)
Prove that Euler's rule, V + R = E + 2, is true for all planar graphs
Even Perfect Numbers      Problem ID: 326 (26 Jun 2007)
Prove that P is an even perfect number iff it is of the form $2^{n-1}(2^n - 1)$ where $2^n - 1$ is prime.
Even Sum Of Two Abundant Numbers      Problem ID: 347 (08 Nov 2008)
Prove that every even $n$ ≥ 48 can be written as the sum of two abundant numbers.
Every Primitive Triplet      Problem ID: 302 (02 Jan 2007)
Prove that the given identities will produce every primitive Pythagorean triplet.
Exponential Symmetry      Problem ID: 344 (09 Jul 2008)
Given that $m$ and $n$ are positive integers, solve $m$n = $n$m.
Factorial And Square      Problem ID: 220 (30 Mar 2005)
Solve the equation a!b! = a! + b! + c2
Factorial Equation      Problem ID: 216 (09 Mar 2005)
Solve the factorial equation a!b! = a! + b! + c!
Farey Sequence      Problem ID: 181 (Oct 2004)
What fraction is the immediate predecessor of 2/5 in F100 of the Farey sequence?
Fibonacci Sequence      Problem ID: 135 (Nov 2003)
Prove that the given nth term formula for the Fibonacci sequence is true
Finishing With 99      Problem ID: 141 (Dec 2003)
Prove that 99n ends in 99 for odd n
Firework Rocket      Problem ID: 161 (Mar 2004)
How long does the rocket take to reach its maximum height?
Fourth Power Plus Four Prime      Problem ID: 355 (23 Jul 2009)
Given that $n$ is a positive integer, when is $n$4 + 4 prime?
General Factorial      Problem ID: 251 (02 Dec 2005)
Show that the Gamma function is a suitable extension of the factorial function
Geometric Division      Problem ID: 304 (12 Jan 2007)
Prove that the quotient and remainder for any prime cannot be in a geometric sequence.
Harmonic Sum Approximation      Problem ID: 209 (17 Feb 2005)
Estimate the sum of the first one hundred Harmonic numbers
Impossible Quadratic      Problem ID: 244 (19 Oct 2005)
Show that px2qx + q = 0 has no rational solutions
Infinite Circles      Problem ID: 367 (15 Nov 2009)
What fraction of the large red circle do the infinite set of smaller circles represent?
Integer Integral      Problem ID: 148 (Jan 2004)
Integrate the integer part function of 10x
Irrationality Of E      Problem ID: 377 (17 Oct 2010)
Prove that $e$ is irrational.
Irrationality Of Pi      Problem ID: 371 (24 Dec 2009)
Prove that π is irrational.
Irrational Cosine      Problem ID: 280 (13 May 2006)
Prove that $\cos(1^o)$ is irrational.
Loan Repayments      Problem ID: 260 (09 Jan 2006)
Find the fixed monthly payments to repay a loan of £2000 in thirty-six months at an annual rate of 12%
Lucky Dip      Problem ID: 276 (21 Apr 2006)
Find the expected return on this game of chance.
Maximised Box      Problem ID: 121 (May 2003)
Prove that the volume of an open-top box is maximised iff the area of the base is equal to the area of the four sides.
Maximum Product      Problem ID: 334 (19 Nov 2007)
Investigate sets for which the sum of elements is fixed and the product is maximised.
Napoleon Triangle      Problem ID: 313 (15 Feb 2007)
Prove that the centres joining the constructed equilateral triangles from any triangle always form an equilateral triangle.
Never Decreasing Digits      Problem ID: 263 (29 Jan 2006)
How many numbers below one million have increasing digits?
Never Prime      Problem ID: 235 (31 Jul 2005)
Prove that 14n + 11 is never prime
Odd Perfect Numbers      Problem ID: 328 (05 Jul 2007)
Prove that an odd perfect number must be of the form $c$2 $q$4$k$+1 where $q$ ≡ 1 mod 4 is prime.
Order Of A Prime      Problem ID: 288 (09 Sep 2006)
Prove that $p - 1$ is always a multiple of $ord(a, p)$.
Pairwise Products      Problem ID: 224 (24 May 2005)
Prove that the sum of pairwise products for a set of three real numbers whose sum is one cannot exceed one third
Perfect Digit      Problem ID: 331 (29 Aug 2007)
Prove that the last digit of an even perfect number will be 6 or 8.
Perfect Power Sum      Problem ID: 307 (20 Jan 2007)
Show that $x$$n$ + $y$$n$ = $p$ has no solution if $n$ contains an odd factor greater than one.
Perpendicular Medians      Problem ID: 296 (10 Dec 2006)
Investigating the properties of a triangle for which two of its medians are perpendicular.
Prime Partner      Problem ID: 291 (22 Sep 2006)
Prove that for every odd prime, $p$, there exists a unique positive integer, $n$, such that $n^2 + np$ is a perfect square.
Prime Reciprocals      Problem ID: 238 (02 Aug 2005)
Prove that there exist no set of primes for which the sum of reciprocals is integer
Primitive Pythagorean Triplets      Problem ID: 205 (24 Jan 2005)
Prove that the given matrix transformation will transform a primitive Pythagorean triplet into a new one.
Quadratic Circle      Problem ID: 168 (Apr 2004)
Find the co-ordinate of the point on the y-axis generated by the quadratic and the circle
Quadratic Differences      Problem ID: 295 (26 Nov 2006)
Determining the values of $n$ for which the quadratic equation has no solutions.
Quarter Circles      Problem ID: 142 (Dec 2003)
Find the area of the shaded region generated by four overlapping quarter circles
Radical Convergence      Problem ID: 257 (01 Jan 2006)
Prove that the Pell equation $x^2 - dy^2 = 1$ can be used to find convergents for $\sqrt{d}$
Random Chords      Problem ID: 195 (21 Dec 2004)
Determine the probability that all r random chords are non-intersecting.
Reciprocal Radical Sum      Problem ID: 266 (05 Feb 2006)
When is the sum of reciprocal square roots rational?
Reciprocal Sum      Problem ID: 169 (Apr 2004)
For a given x, determine the number of solutions of 1/x+1/y=1/z
Repunit Divisibility      Problem ID: 293 (03 Oct 2006)
If $GCD(n, 10) = 1$, prove that there exists some repunit which is divisible by $n$.
Sliding Box      Problem ID: 155 (Feb 2004)
Can you find the coefficient of friction between the two surfaces?
Square Lattice Triangles      Problem ID: 162 (Mar 2004)
Find the probability that three randomly chosen points of a 5 by 5 lattice will form a triangle
Square Sum Is Square      Problem ID: 336 (27 Mar 2008)
Prove that there exists a sum of $n$ distinct squares that is also square.
String Of Ones And Zeroes      Problem ID: 187 (01 Nov 2004)
Prove that there always exists a number made up of ones and zeroes that is divisible by the positive integer, n
Sum Of Two Abundant Numbers      Problem ID: 348 (08 Nov 2008)
Prove that all integers greater than 28123 can be written as the sum of two abundant numbers.
Tangent Sum And Product      Problem ID: 115 (Apr 2003)
Can you prove that the sum of the tangents is equal to the product?
Time Loses Integrity      Problem ID: 272 (16 Feb 2006)
Prove that the time the body takes to pass between two point can never be integer.
Unique Square Sum      Problem ID: 176 (May 2004)
Prove that if a prime is expressible as the sum of two squares it can be done in only one way