Spectrum Agency Test

37 problems

Time: 61 minutes

Questions

(1) You have to take three criminals back to the County Jail: Alexei Asimov, Carlo Carlucci and Walter Trunch. You have to cross a river on the way, and the boat only takes two people at a time. The trouble is that if you leave the criminals together, Asimov will kill Trunch, and Trunch will kill Carlucci. How can you get them safely across the river?

(2) You have seven gold bars. However one of them is a counterfeit and weighs less than the others. You have a set of balance scales but you may only use them twice. How do you identify the counterfeit gold bar? 

(3) Calculate:

(4) The entrance to enemy HQ is protected by a special lock. To break in you must arrange the numbers from 1 to 9 in a 3x3 square so that if you add up the numbers in any row, column or along the two diagonals the answers are all different.

(5) You are a Spectrum agent in space and you have an important mission to visit 9 space stations. However after lift-off you discover a glitch in your rocket boosters. The space craft will only travel in straight lines and you only have enough energy to fire the boosters four times, to make four changes of direction. Can you complete your mission?

(6) To break into enemy HQ you have to enter a number into the keypad. You’ve been tipped off that the key is given by the next number in the following sequence:1, 11, 21, 1211…What is the key?

(7) The precious emerald of Copan was first discovered in 25BC but it went missing 50 years later and has never been since. What year did the emerald disappear?

(8) You’ve been observing the entrance to an enemy missile launch site trying to crack the secret to getting in. A scientist turns up at the gate. The guard asks “Twelve?” and the scientist replies “six” and is let in. A second scientist arrives. This time the guard asks: “Six?” The second scientist replies “Three” and is let in. You think you’ve cracked it! So when you approach the gate and the guard asks “Nine?” why did the guard sound the alarm when you answered “four and a half”?

(9) There is a meeting for the 9 new agents who have joined Spectrum. How many handshakes will it take for all the agents to meet each other?

(10) You are out on a mission and need money to buy supplies. But when you take out £100 from the Spectrum account you are told that the money left in the account has doubled. How much was in the account before you withdrew the money?

(11) You have discovered a U-shaped detonator. In order to defuse the bomb you must cut the detonator so that ALL the connections are cut. But you only have time to make 2 straight cuts before the bomb will explode. How do you defuse the bomb?

(12) Spectrum has nine cells arranged in a 3x3 square with connecting windows. You’ve captured three gangs of three criminals: the leader, second and third in command of the red, yellow and blue gang. You have to arrange the prisoners so that no one from the same gang can see each other and also so that no one of the same rank can see each other. Call the prisoners from the red gang R1, R2, R3, from the yellow gang Y1, Y2, Y3 and the blue gang B1, B2, B3.

(13) A spectrum agent travels to work on his motorbike at 30mph. On his return journey, the roads are clearer so he travels at 45 mph. What is his average speed?

(14) Spectrum HQ has been surrounded and there is no way to bring in more food. The trapped agents will have to survive on what they’ve got inside. 40 agents would have enough food for 20 days. 30 agents would have enough food for 30 days. How long will the food sustain 25 agents?

(15) Spectrum are holding a knock-out tennis tournament. There are 81 agents who enter. How many matches do you need to organize to find the champion?

(16) You are out on a mission. You travel 5 km south, then 5 km west, then 5 km north and find yourself back where you started! On your mission you saw a bear. What colour was the bear?

(17) Count von Count has concocted an evil plan to take over the world. Every month each member of his gang must recruit a new member. So each month the gang doubles in size. How long would it take for the whole population of the world to be in Count von Count’s gang? 

(18) You are guarding Spectrum HQ and your shift begins at 12 midnight and finishes at 12 midnight the following day. You spend a lot of time looking at your watch. How many times does the minute hand pass over the hour hand? 

(19) You are in the basement of Spectrum HQ. In front of you are three light switches, all in their OFF position. One of the switches is connected to a lightbulb in the room at the top of the building but you don’t know which one. There’s no way you can check, unless you climb the stairs. You have as much time as you need to play with the switches in the basement, but you can only go upstairs once to check on the state of the bulb. How can you tell with 100% certainty which switch operates the bulb?

(20) You and fellow agent Bob Black have been captured by the enemy. You are told that you are going to be blindfolded and taken into a room where red or green hats are put on your heads. The blindfold will be removed and then you have to write down on a piece of paper (without talking to each other) the colour of your own hat. If any of you gets his colour correct he can go free. Can you devise a strategy with your fellow agent that will guarantee at least one of you goes free?

(21) You have infiltrated a criminal gang that has stolen 100 diamonds from the state jewels. There are ten member of the gang including you. Each gang member can propose a way to divide the diamonds and a vote will be taken whether to accept the proposal. As you are the newest member of the gang you are asked to go first. The proposal is accepted if half the members or more vote for your proposal. (You’re allowed to vote too). Otherwise you will be shot and it’s the second most junior gang member who goes next. If he loses the vote, he dies and it moves to Gang member number 3, and so on until a Gang member wins the vote. What is your best strategy? You certainly want to survive but you also want to take back as much of the stolen diamonds as you can. Remember these gang members are very clever, very greedy and none of them wants to die.

(22) Spectrum have sent you a new credit card but in order to find out the PIN you have to crack the following mathematical puzzle: the PIN is the smallest number that can be written as the sum of two cube numbers in two different ways.

(23) It appears that Agent P might be working for the enemy. But in order to get into the safe where he keeps his private documents, you have to enter a string of 8 numbers. You’ve got 7 of them but what’s the last number that will complete this sequence 13, 17, 31, 37, 71, 73, 79…?

(24) You have been caught by the enemy and trapped in a collection of room full of doors. To free yourself you have to find a way to get to the exit by going through all the doors once and once only. If you go through a door twice the building blows up.

(25) Spectrum agents Bret and Emily and Chuck are all driving to a Clairvoyant’s concert. They set off an hour before the concert. Bret takes route A which is twice as long as Emily’s route B but the average of their two routes is the same as Chuck’s route C. Trouble is Bret gets lost and goes 10 miles out of his way, meaning he ends up traveling as much as the combined distance Chuck and Emily travel. Assuming they all drive at 40 miles an hour, how late is Brett for the concert?

(26) In the Spectrum canteen today’s special dish of the day has got a special price: “Cryptic Curry costs £2 plus half its price”. How much does Cryptic Curry cost?

(27) You are on a mission to the snowy Alps. Your skis are 5 feet long but the Spectrum jet that is due to take you to the Alps does not allow any luggage that is over 4 feet by 3 feet so the pilot refuses to take you. How can you get to the mission on time?

(28) You have to cross a crocodile-infested river. There are 40 stepping stones that you can jump across to get to the other side, but you notice that there are crocodiles next to every sixth stone. You can jump up to 10 stones ahead at a time, but you must jump the same number of stones ahead with each jump. How many stones should you jump each time to get to the other side without being eaten by a crocodile?

(29) You are trying to break into an enemy agent’s apartment, but to open the door you must place a hexagon against the entry pad. However, all you have is a cube. How can you cut a cube with one cut to reveal a hexagonal face that you can use to break into the apartment?

(30) An enemy agent has just jumped on board a moving freight train. The train is one mile long and is travelling at 15 miles an hour. It is just about to enter a tunnel which is also one mile long. You need to get to the other end of the tunnel before the whole train has passed through. How long does it take for the train to pass completely through the tunnel?

(31) You are on a mission with three Spectrum agents and a sniffer dog. You reach a river, but the boat to cross it can only take a maximum of 150 pounds in weight. You and agent X weigh 75 pounds but agents Y and Z each weigh 150 pounds. How many trips will it take to get all the agents (and the dog) across the river?

(32) While on a mission in a forest you realise that the enemy have set fire to the forest and it is spreading rapidly in your direction, aided by a strong westerly wind. You have no transport and no water to put the fire out. All you have is box of matches. How can you save yourself from the approaching fire?

(33) What’s the next symbol in this sequence?

Spectrum Agency Test

Solutions

(1) Solution: First take Trunch across the river. Asimov and Carlucci are safe together. Then take Asimov across but since you can’t leave Asimov and Trunch together you must bring Trunch back. Then take Carlucci across the river. Finally come back for Trunch. Everyone is safely on the other side.

(2) Solution: Put three bars on each side of the scales. If they balance then the gold bar you didn’t put on the scales is the counterfeit. Otherwise take the three bars that are lightest on the scales. Take two of these bars and put them on either side of the scales. If they balance, then the bar you didn’t put on the scales is the counterfeit. Otherwise, the lighter of the two bars is the counterfeit.

(3) Solution: 42

(4) Solution: This is the opposite to a magic square where all the rows, columns and diagonals add up to the same number. There are actually lots of solutions (24960 to be precise). Here’s a nice one where you arrange the other numbers in increasing order in a spiral starting with 1 in the middle:

(5) Solution:

(6) Solution: 111221. Each number describes the previous number. One; one “one”; two “ones”; one “two” and two “ones”.

(7) Solution: Most people’s response is 25AD but that is incorrect! There was no year 0 so actually the emerald went missing in 26AD.

(8) Solution: You should have answered “four”, the number of letters in the question.

(9) Solution: Agent 1 shakes 8 agents’ hands. Agent 2 has already shaken hands with agent 1 so only has to shake 7 more hands. Continuing like this we need 8+7+6+5+4+3+2+1= (8+1)+(7+2)+(6+3)+(5+4)=4x9=36 handshakes.

(10) Solution: the account was £100 overdrawn. When you withdrew £100 it became £200 overdrawn: (-100)x2=-200.

(11) Solution: First cut along line AB. Then put the three pieces together such that lines CD and EF line up and you can cut along the combined line in a single cut.

(12) Solution: this is one possible way to arrange the prisoners:

(13) Solution: 36 miles per hour. Most people think the answer is 37.5 miles per hour – an answer obtained by adding the speeds together, 30+45=75 – and then dividing in by two, to obtain a simple average. But that is wrong.Speed is actually worked out through a simple formula – distance divided by time. So suppose the distance from the agent’s home to Spectrum HQ is 45 miles. This would take 1.5 hours in the morning and 1 hour on the way home. The average speed would then be distance – 45+45=90miles – divided by time – 1+1.5=2.5hours. This gives us the answer 90/2.5=36 miles per hour. You get the same answer however far the distance is between Spectrum HQ and the agent’s home.

(14) Solution: 40 days. Many people think the answer is 35, but that’s because they are assuming that there is a direct relation between agents and days. But actually it is an inverse relation. The more agents you put in, the less days the food survives. This means that if you plot a graph with number of agents on one axis and the inverse of the days on the other, you will get a straight line. (The inverse of a number means you calculate 1 divided by that number: the inverse of 20 is 1/20.) We can mark two points on the graph: the first point is at 40 agents and 1/20 for the inverse of the days; the second point is at 30 agents and 1/30 days. Draw a straight line through these two points. How high is the graph above 25 agents? The answer is 1/40. So that means if there are 25 agents then the food inside Spectrum HQ would last for 40 days.

(15) Solution: 80 matches. This is because each agent loses once – except for the champion.

(16) Solution: White. The only place on the earth where you can go south 5 km then west 5km then north 5km and end up where you started is the north pole. When you go south you are on a line of longitude. Going west you run along a line of latitude. Then heading north you again drop down a line of longitude. The only bears at the north pole are polar bears. So the bear was white.

(17) Solution: Only 33 months. Each month the gang doubles in size. So after one month there are 2 gang members, after two months there are 4 members, after 3 months there are 8 members. If you keep on doubling then after only 33 months there are already over 8 billion gang members. The population of the earth is approximately 6.7 billion. Doubling makes things very big very quickly.

(18) Solution: 21 times. Between 12 midnight and 1am the hands don’t cross. Then between 1am and 2am the hands cross once, just after 1:05. During each hour the hands cross once until 11am. So that’s 10 times. The slightly unexpected thing is that between 11 o’clock and 1 o’clock the minute hand passes over the hour hand only once – at midday. From 1pm to 11pm the hands again cross 10 times. Then in the final hour they don’t cross. So that’s a total of 21 times.

(19) Solution: Turn switch number 1 and 2 ON. After a few minutes turn OFF switch number 2. Now go upstairs.If the bulb is ON, it’s switch number 1If the bulb is OFF but WARM, switch number 2 operates it.If the bulb is OFF and COLD, it’s switch number 3.

(20) Solution: You write down the colour of Agent Bob’s hat. Agent Bob writes down the opposite of your hat. Both of you can’t both be

wrong. (You also can’t both be right). So one agent exactly goes free.

(21) Solution: You just need four gang members to vote for you since including your vote that will be 5 votes to 5 and the proposal is passed. So you propose to give Gang members 3, 5, 7 and 9 one diamond each and you keep the other 96 diamonds. Why do you get the vote of Gang members 3, 5, 7 and 9? Because they realise that unless they vote for you, they will end up getting nothing or being shot. For example, imagine there were just three gang members. You offer gang member 3 one diamond. He will vote for you because otherwise you get shot, and then Gang member 2 proposes to keep the lot, votes for himself and gives Gang member 3 nothing.

(22) Solution: 1729. Because 1729=123+13=103+93.

(23) Solution: 97. Agent P is a big fan of prime numbers, numbers that are only divisible by themselves and 1. The sequence of numbers in his safe code is made up of the reversible primes, prime numbers that give another prime number when you reverse their digits.

(24) Solution:

(25) Solution: fifteen minutes.

(26) Solution: £4. Many people think it’s £3 but that’s not right. If curry costs C pounds then C = 2 + C/2. Put C=4 and the equation is right: 2+4/2 =2+2=4.

(27) Solution: Get a 4 foot by 3 foot rectangular box and put the 5 foot skies diagonally inside the box. Who said Pythagoras wasn’t useful!

(28) Solution: 7 steps ahead. The first number divisible by both 6 and 7 is 42 so you will get to the other side without meeting a crocodile. All the other choices fail. For example, if you jump 9 steps ahead then you will meet a crocodile at the 18th stone.

(29) Solution: Imagine hanging the cube from one of the corners. If you cut the cube horizontally between the point at the top and point

cuts all six faces. Each face contributes an edge to the new face. The shape should be symmetrical. Hence you get a hexagon.

(30) Solution: 8 minutes. When the front end of the train enters the tunnel, the whole length of the train is still outside the tunnel. After the train travels one mile, the front end leaves the tunnel, but the whole length of the train is inside the tunnel. After it travels a second mile the rear end of the train leaves the tunnel. So the time it takes to pass through the tunnel is the time it takes to travel 2 miles. The train does 1 mile in 4 minutes so the total time to pass through the tunnel is 8 minutes.

(31) Solution: 11. First you and agent X cross. You return. Agent Y crosses. Agent X returns. Then both you and agent X cross again. You return. Agent Z crosses. Agent X returns. Both you and agent X cross again. You return. You and the dog cross.

(32) Solution: Use the matches to set fire to the ground where you are. Then walk west towards the main fire. The wind will blow your fire further east, thus burning out the land to the east of you. Then walk back to the burnt area of land as the main fire approaches. It is impossible for the fire to catch light here, as there is nothing left to burn.

(33) Solution:

It’s the numbers next to their mirror images at the bottom you have two pieces each with a new face. The slice